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MATHEMATICS LIBRARY 
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The 
Frank Hall collection 
of arithmetics, 
presented by Professor 
H. L. Rietz of the 
University of Iowa. 


OFO6H 513 


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UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 


AUG 2 4 1999 
SEP 1 4 ECD 


L161—O-1096 


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Sen» 


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MATHEMATICS FOR CoMMON SCHOOLS 


ARE eh 


GRAMMAR-SCHOOL ARITHMETIC 


INCLUDING 


EASY ALGEBRAIC EQUATIONS AND SIMPLE 
) GEOMETRICAL PROBLEMS 


BY 


JOHN H. WALSH 


ASSOCIATE SUPERINTENDENT OF PUBLIC INSTRUCTION, 
BROOKLYN, N.Y. 


BOSTON, U.S.A. 
D. C. HEATH & CO., PUBLISHERS 
1897 


CoPpYRIGHT, 1895, 
By JOHN H. WALSH. 


Norwood J8ress : 
Jj. S. Cushing & Co. — Berwick & Smith. 
Boston, Mass., U.S.A. 


oy eg a cnet ie 
W \6-m marnennarics UA 
| es 


PREFACE. 


THE Primary ARITHMETIC, Part I., of the two-part edition of MATHE- 
MATICS FOR Common ScuHoots is designed to cover the work of the first four 
years, and contains those portions of the subject needed by all pupils of the 
common schools: addition, subtraction, multiplication, and division of 
whole numbers; simple fractions; and the most commonly used denomina- 
tions of compound numbers. 

Part II., the GramMMAR ScHoot ARITHMETIC, completes the ordinary 
grammar-school course in this subject, and contains, besides, two chapters 
on algebraic equations and one on elementary constructive geometry, with 
applications. The first algebra chapter should be taken up with the 
seventh year’s work in percentage and interest. The remaining one, 
Chapter XV., may be profitably studied where there is time to continue 
this subject. Although placed at the end of the book, it is intended that 
suitable portions of the geometry work of Chapter XVI. be taught from 
time to time during the last two years of the grammar school. 

The special features of the work are its division of the arithmetical 
portion into half-yearly chapters, instead of the ordinary arrangement by 
topics; the omission, as far as possible, of rules and definitions; the very 
great number and variety of the examples; the use of the equation in the 
solution of arithmetical problems, especially in those of percentage and 
interest; and the introduction of the elements of algebra and geometry. 

Believing that there is some foundation for the complaints frequently 
made by business men and high-school teachers that grammar-school gradu- 
_ ates are too often slow and inaccurate in ordinary computations, the author 
has furnished throughout the entire work systematic drills and reviews in 
the addition, subtraction, multiplication, and division of ordinary numbers 


and of fractions. 


464193 


lv ‘PREFACE. 


In this endeavor to enrich the grammar-school course in mathematics, 
the attempt has not been made to shorten it so much as some may desire. 
The intelligent teacher can and should do the remainder for himself, by 
rigorously omitting all such topics as he finds unnecessary. 


J. H. W. 
Brooktyy, N.Y., January, 1895. 


CONTENTS. — PART II. 


pte a 
CHAPTER VI. 

PAGE 

Mrxep NumBrers— FEDERAL Monry — Britis — DENoMINATE Num- 
BERS — DECIMALS — — MEASUREMENTS GUNN es AMG = staal oi, Lehane keen 
Mixep NuMBERS. . . oy Va Ang Ae NE irint TA Waly ene inh aaah NE 
Addition of Mixed Re PEC ERD ARR eR i uti No M8 
ME UPMITOSPATIC TE CUTE trannies va ics Srt Wiel heat tyes Nk ake ht etna 
PDEPACuOt OL) WiIXOG) INUIMUETS Noh, Huretrees wr seh ihe tet al, on fea inal Reo 
PrestNman DE NUMEEATION( +) 4G Vict bestal ve Ble) lc avy et ep iehe ae 
Niinucanomot Mixed Numbers (5 save) ei oy ist) s hos «) em ete 
Division-of Mixed Numbers.) v2 ee a4 
SES Ee ALON Ne MEET oedine Masitinge oie) hae Urar Uwat) 9tat aude unter 
Pracuorabe arts Ol SOU ET i". ue bse) edie Agi cn s,m aah 
Dire OO STA MONGY Cree and ccs Pine Mi) cat abuses oso veer entre 
Pere DECK IMALION A 4) Wall Vaio ils cet orn WAN Lowe, elt he iterabe at taal 
PENA TH ANU MEMES mre oy cums el | aiidt piece Gerd ey les yee aoe eee 
Ue VIDOES a Ree cls A i eee et OT Ai VR sage 
PRE NOC HSTIR Gum Aries Weta alts. | Miia ie Latics’ Eile ya. es sl) a oe engl 
PUR IECITSOAR LV CLOD Laie is ce Ye siGh, iei\ col Watt ve Mev lan Va aati os See a Rene 
Breer MN HGRULO Mme eater Myo Uw wr er Sud Pratl) ch atG Lega vats eee 
CATE VEE Bg VEE Gof pee a Be TOR ny ora ica St ec eae nm a) 
SENG og wake hy SST AM AR TTR Sy ce got ae 
SRE RIES DE a OR el My et tae ee SW) ot dae ian Ng, le Al WY cite MER 
POMRLROT CA TCS NUMIGL ALON 9 0. Jill’. sical es Rae Sac cay ble. lm. 1s ne eae AL) 
Pe RTCPRDAOT COR Ori Are oss Vale ag aed MEL Pate Hays kek del o'r Sete 
Subtraction of Decimals . .. . ROL Ppt ia OO) nanan Tee MANN PS 
Multiplication of a Decimal by an Tee! WON e SU sar, rap ete Pan Nel Gib? 8, 


Dimon. cis Decimal by an Integer... ew es [5 Ne eo) B44 
Aan eT eee ee rae Sipe nS. CEN eh, tatty A aot went BAB 


al CONTENTS. — PART II. 


CHAPTER VII. 


Fractions — Decimats — Brnits — DENoMrINATE Numpers — MEas- pier 
UREMENTS: 5 ei ne ahh) peg ety, RCNP Og 
ADDITION OF FRACTIONS... 0. Una ewe ae ee 
SUBTRACTION OF IPRAOTIONS “4.0. ac a). 86 4 Ny 
Factors and Multiples’. ..(7 3) RAL 0%) oa ke 
Prime Namberp eo ss oa ake Gl eee 
Greatest Comnion Divisor .. 4) 4 6 Ga wh le ee 
Lowesb'Terms 20 4h awe ce le) ay cs 

Jieast’' Common Multiple’ 2. 5 2k ee 8 
ADDITION AND SUBTRACTION OF FRACTIONS: (34/04), <9) oy oe 
Special Drills oy Sl he ey ce ee a 
Cancellation. ive ce. le ye ee eta 
MULTIPLICATION OF (EP RAOTIONS «oA. 6) isha as eee 
DIVISION OF PRAOTIONS «S/S 5, tees. ee ticey © ok od 
Fractional Parta of a: Dollar vi). 3.) ee Ve 8.) ee 
BrByS hoops ha) eo tary 8) aloe 561 ween el eG Ok 9 eee 
Short: Methods) (60 eg eee alle eg ol) a 
MOLTIPLIGATION OF DECIMALS 40: ss 5s ee et 
DIVISION: OF DEOIMAIS ie RP ge hay ek a tee ne 
Sight Approximations: ; (5 )...0!s) 3/5 0) ee 
DENOMINATE: NUMBERS «0.5/8 0k Gd 
ong’ Measure. hse bee er tt it ay a, 
MBASUREMENTS © 481 ae RS he ee Ol ke, nent 


CHAPTER VIII. 


DECIMALS — Brits — DENoMINATE NumBERS — MEASUREMENTS — 


PERCENTAGE — INTEREST’ =: 2)... of 3's eS Se 
DECIMAES) 0 Ry ae ns a gee ke: | bela a ata 
Reduction “0 eee oy a) oo Sys ea te 
Addition’ ai poe: ae i he len ge) 
Subtraction. 2) is) faa en) co al ce 293 


Multiplication’: 2% 0.0.0) Whoa) et ns tice e) « Jira rtnnt re ee 
DiVIRIOD 6.556 ye SE, ee dag eee ae 


CONTENTS. — PART II. 


MEASUREMENTS 
Special Drills . 
Short Methods 
Approximations . 
DENOMINATE NUMBERS . 
Reduction Descending . 
Reduction Ascending 
Addition and Subtraction . 
Multiplication and Division . 
PERCENTAGE 
BILLs . 
INTEREST . la, 
AREAS OF Ricut-ANGLED TRIANGLES 
Short Methods 


CHAPTER IX. 


DENOMINATE NuMBERS— SURFACES AND VOLUMES — PERCENTAGE — 


INTEREST 
DENOMINATE NUMBERS . Senne 
Reduction Ascending and Descending 
Compound Addition 
Compound Subtraction . 
Compound Multiplication . 
Compound Division . 
Special Drills . 
Short Methods aetie 
Avoirdupois Weight (Long Ton) 
MEASUREMENTS 
Time between Dates 
PERCENTAGE 
INTEREST . 
Approximations . 


vil 
PAGE 
297 
298 
303 
305 
307 
311 
311 
312 
314 
316 
319 
320 
322 
323 


326 
326 
326 
329 
331 
332 
333 
337 
340 
341 
342 
345 
349 
349 
303 


Vill CONTENTS. — PART II. 


SURFACES 
Square Measure . 
VOLUMES : 
Approximations . 
Cubic Measure 
Troy Weight . SURO AER UF 
ANGLES, TRIANGLES, QUADRILATERALS . 


Areas 
CHAPTER X. 
ALGEBRAIC EQUATIONS. 
OnE UNKNOWN QUANTITY 
Clearing of Fractions 
Transposing 
CHAPTER. XI. 


PERCENTAGE — INTEREST — DiIscouNT— SURFACES AND VOLUMES . 


PERCENTAGE free ena 
To find the Base or the Rate 
Profit and Loss 
MEASUREMENTS 
INTEREST asthe 
Interest-bearing Notes . 
Special Drills . 
Approximations . 
Short Methods 
Bank Discount iE aE ee 
Discount of Interest-bearing Notes 
English Money . 
CoMMERCIAL DIscouNT . 
SURFACES AND VOLUMES . 


CHAPTER XII. 


SIMPLE AND CompounpD INTEREST — Discount — CAUSE AND EFFECT — 
PARTNERSHIP — Bonps AND Stocks — ExcHANGE — LONGITUDE 


AND TIME— SURFACES AND VOLUMES . 


PAGE 


353 
354 
358 
360 
361 
361 
365 
367 


069 
369 
373 
377 


381 
381 
383 
386 
389 
393 
394 
396 
398 
399 
402 
407 
408 
410 
413 


415 


CONTENTS. — PART ITI. 1X 


SimPLE INTEREST. . . . LSBs iy HEM Mal ay) «| Cae EO 
To find Principal, Rate, or Time SOV ies h ceed Coe ReeMeae SAN ge Ley 
interes py Asiquot Farts 12 Teeny. Neer tee shee CATT 


DIE METER ML IRCOUNT. Poa Sets cs Leal, ele ieneatine Liduhielus \aelttern tes bak 
BANE DISCOUNT». <,' Sk eC edey e ea tel eeLe 
To find Face of Note, Bite ae Diaeoune or Time iteatt Fe) cia. NBS Ml ene ree 
SUS EU MAYTALS aR aL thea? ha re aad aC A ei a AT 0 

BEES MOUNOCS Ree ete en Lah eR R ESR ies ocean UN ae 

MARR RSET ee, Gh al ao te Wage Mee Me RI RACY A 
TMT GROTON et iia ey Coe ih ay ce Macdkye Mra Satter) atinte sa uin ere ey 
CEL Pang TES 2 A PL PT Sg SO A POLAR es 2 
Pe POEUN AON con Wine Mim eM re gh tes Meus nant ail ound Aen ng Pet nee 
PRR EPEC STOOLS) oh e Yshecte cme Ch cf SME ate (at ec a. caiiita ine ee ee ae 
Beem UMTS INTEREST eh. 5 fist tes) ls ai) co Mtlpenie es ES A Ne A ate 3 SBO 
EXCHANGE... POEL RES oy SUI aN Mate RIN Oat HAN VEE eae hue 
Domestic Sight vee RA Keok OHMS GI SNe Pela yenreane pa 
CHPCULAT? WLGABUNO ror) Sue cia ee roe hie Lah ape tar “aytanmieerL cmap OL) 


EGET IEALCAPET IIIS. chet eM ic) san te fauitica Is Welple W alata eer) woah y 
LoNGITUDE AND TIME. . . SERN eatin te ch cal ta A sep sat Aveda gm WML EDD 
Bills of Exchange Feathany BM Teme e Neh NC at! onl! Riera!) Sula es 


CHAPTER XIII. 


PARTIAL PAYMENTS—RATIO AND PROPORTION —SQuARE Root— 


RRDURFACES: AND) VOLUMES. Bee ppd eel Ghia 2 hs 458 
eatTA PAY M ERTS UO ULE aes) Oe Pe Alay o) KAU 488 
Presentaworou and True’ Discounts 67. 2) eee eo. 460 
ene eee GVOLOMES Ie. Wt seh eh ay tn) RO wy 4OR 
Der H aTCO Teme terete isha Coy Nits 1) MOP eS ae UE Oa 
TRA TIO cha sene Mk, Pah ae ere ANI IG) eh Watts A RE EOS 
Special Drills EA a URS DRA RAB URS URC MCS RW DRL Be Yi 
PROPORTION. . . tga Ve Shae eR ly hie ne Mas es Ch wee 


Applications of ey EOOU MAEM tatu ge lie VR tut cumay Tenis Vas Stat heOo 


x CONTENTS. — PART II. 


MEASUREMENTS: 05's (solu SDR ce a 
Hixact Interest ecg i Picea eet ae 
PARTIAL PAYMENTS— MrRcHANTS RULE... ..... 


CHAPTER XIV. 


HQUATION OF PAYMENTS—MENSURATION oF SURFACES AND Vot- 
umMES — BoArD MEAsuRE— ANNUAL INTEREST — GOVERNMENT 
Lanps— Merrie System 

EQuaTION OF PAYMENTS 

MENSURATION OF PLANE SURFACES 
Special Drills . 

SURFACES OF SOLIDS 

PPISHIS ATIC ULI OOPS 5 one 7 i ae ae a ate 
Pyramids and Cones 

VAT NCIS 100) SET) ST Es SN ak ha ha es ca 

Lumber Measure 


Surface of Sphere 


CORE Ronis oyet bree ide ieee Nak est Galle) WO Ui an an 
Volume ofSphere)( i) Py 0). 4 ld eee ee ae 
ANNUAL LNTER EST Gir ct. (llc evil cunedine, wean Outen Can er 
Government Danda rye) Mle ese ate ene 
RMIETRIOUS YSTEMS 575. ho OUR ae oe AGED oe ee een 
CHAPTER XV. 


ALGEBRAIC Equations—Two Unknown QUANTITIES— THREE Un- 
KNOWN QUANTITIES — PURE QUADRATICS — AFFECTED QUAD- 
RATICS 

ADDITION oF ALGEBRAIC QUANTITIES 

SUBTRACTION OF ALGEBRAIC QUANTITIES 

Removing Parentheses . 
Two UNKNOWN QUANTITIES . 
THREE UNKNOWN QUANTITIES 2; . 3 wh 6 ue 


499 
499 
503 
505 
511 
511 
512 
513 
514 
518 
519 
521 
523 
524 
525 


530 
530 
532 
534 
538 
544 


CONTENTS. — PART II. Xl 


PAGE 
MULTIPLICATION OF ALGEBRAIC QUANTITIES . ........ 5A 
ROPER EGIL A UA TEC cee c! gle’ (ass etd MMOH MO UO eer a AC Gis POLO 
PUR PROTEINS AIGA DATION 9). Seu Wee ha ee) Leica AG Hat Py BBL 


CHAPTER XVI. 


ELEMENTARY GEOMETRY — PROBLEMS IN CONSTRUCTION — PRACTICAL 
_APPLICATIONS—CALCULATION OF HEIGHTS AND DIstTancES— 


BU TT O Mere fee ism bettie vise TAN Sieg eis Lo eat area EEN 
Pee CA LT BOM ET VG te ells SG eck be Wes Ugh gel Mgl Yene lhe Fe cayh OOy 
Eee EAU ITLA ETUC OID tins (cl eo an hls doled iar eae cede ahr ty ane 
PCE CC MI EUPUCHOT ear cae ee ee lat eno fa ieee EO 
Equal Triangles— Equivalent Triangles . ...... =. =. 584 
MLC AUP lGM ew Aah lb ada a: Beams hirer tates “ky eh nat ee OOO 
PALQuuATION OF HEIGHTS AND DISTANCES... 8. 0. 6 6 ne 1087 
Mee ere TOME OP EOUREAUES (0/200 sila (le. uids ah cel Meyiet oo ele le Dee 


Prams, Ovinderss Pyramids; Cones ah) 6) ss Ne sce, ee yn ia a OOS 


Frustum of Pyramid or Cone ..... . CLUE CNP SAAN it Sie, 8: 
Sirs meM MMP EN Laerir is cea (Ce Mt fal! all aioe se 4 tere AOS 
Stee ROM ASM Sac esi eh taf al Ls she) aie, oan, fer) ce ahve nel BO 
Breer UU OOTS ie Mrs eae hel at hd: tel fU pit wi Sin iin ell eh eig Mae 
Bye PDL STAT CML ORIN AEs Wig Tru'gic fs Wibeh Gea iier he lite) (#i-s)) ay Koay sh OMD 
Brmecama Of byramiag and Cones’. 0. a a Me week ot le ey LOOL 
ReeeVUL Ge Lips mens uae Ge wel e ag) ve tye alcUycay Tile) Tg tahh tet: baanl (<6 Liat IO 
Bere ms Era hte hah ho A Ant igh a Val a Winmteh S) # 7! ool CODE 
APPENDIX. 


iene ome W RIGHTS AND MEASUBES - lu pote ts ie 8 ei ce ees BOF 
PERC neo Lens Wee) RON) eve nin se hls Tarr Cre a BOO 
Days OF GRACE . . . . . . . . . . e . . . . e . . ° 609 


Xi ‘CONTENTS. — PART II. 


PAGE 

RATES OF INTEREST °° ce 8)0) hi EA Es Ee eed ee 
PARTIAL Payments, Connecticut Rule. 2) o>. (cee 
Annual Interest: Notes)... oS ER i ee ee 


New: Hampshire Rule. .)5 ea ear. ote 
Vermont Rule!) \ oi apes Sree ele te eumen ee 
TAXES wel EE Woon) Met dM ial NS Tin eer tra ah Ae, Ar 
Vermont Method of Levying Taxes). 3 5 eee 
VaLuEs Or Forrian Gores) Woe Wie) ao eg area 
} B65). en PE PEERS UPAY RRR GR ie ne gS. 


GRAMMAR-SCHOOL ARITHMETIC. 


CHAPTER VI. 


MIXED NUMBERS. --FEDERAL MONEY. — BILLS. — DENOM- 
INATE NUMBERS. — DECIMALS. — MEASUREMENTS. 


MIXED NUMBERS. 
447. Oral Exercises. 


How many halves in 1? How many fourths in 1? Six 
halves=? 12 fourths =? 6thirds=? 12 sixths=? 


448. Slate Exercises. 


- Add: 

es 2. 58 3. 182 4, 32 5. 744 
18 391 1502 4 34 
274 17 572 272 1. 
449. Oral Exercises, 
$=? $=? $=? §=? Ya? yo? 


450. A mixed number is a whole number and a fraction. 


451. Reduce to a whole number or to a mixed number: 


So re ee an a 


199 


200 ARITHMETIC. 


452. Slate Exercises. 


Add: 
6. 3h Tour 8. 9% 9. 3182 10. 875%, 
95 291 481 53 172 
2544 734 351 527 353, 
74 61 34 14 694 


453. Oral Exercises. 


How many quarts in a gallon? 

What part of a gallon is a quart ? 

4 gallon = how many quarts? += how many fourths? 

How many quarts in a peck? What part of a peck is one 
quart? One-half peck is how many quarts? One-half peck = 
how many eighths? 

+ peck is how many quarts? +=—how many eighths? 2= 
how many eighths? 3= how many eighths? 


coleo 


at = eal ete fa 1 al Ale 
ne? 12 12 12 12 12 12 12 12 1z 12 12 


454. Draw a line one foot long. Draw a second line of the 
same length; divide it into halves. Divide a third line of the 
same length into three equal parts. Divide three other lines, one 
into fourths, one into sixths, and one into twelfths. 


: 
: 


MIXED NUMBERS. 201 


How many inches in a foot? What part of a foot is one inch? 
4 foot = how many inches? 4= how many twelfths? 
= how many twelfths? #—how many twelfths? Change 


1 to twelfths. Change 3, ? ia twelfths. How many twelfths = 
4? 22 82 42 22 8? 


ee eer PETES Sef 4! 
sy ames spears ye deci aris 
Rewer s et? Steet? Sie 
eS Peri Pes Siege Daehn Grea by ones 
10 =? pe EN Tele PE IAG | 
oer oO PE Pe ares | wert Bina - 


How many inches in 3 ft.+ 4 ft.+ 4 ft. +2 ft. +7, ft.? 
How many feet and inches? 

How many 12thsin3+4+41+1+4,45? Change to a mixed 
number. Change the fractional part to a different fraction hav- 
ing the same value. 


What fraction of a dime is 1 cent? 4 dime = how many 
cents? += 75. 

4 dime=how many cents? 4,5. Change #to tenths. 3. 4. 2. 

Add 4 dime, + dime, and ;4; dime. How many cents? How 
many tenths = 4++4-+44,? Can you change the answer to a 
different fraction having the same value? 


455. Slate Exercises, 


Add: 
sb a BS 1262) EG 8 ie os ees os Beige aa Bs 
2704 53h 61 164 1834 
31 9535 172 O54 O1 
16 " eae u uae: 


4.56. Oral Exercises. 


Show by a diagram that 4 is the same as 2. 

How do we add 4 andi? Show by a diagram. 

How many hours ina day? Indday? Iniday? In} 
day? Iniday? Iniday? In +, day? 
Change 1 to oa -fourths. 4. 4 4. 4. 4. 
Reduce 2, $, 8, 2, 3, z5, $, 44 to 24ths. 


202 ARITHMETIC, 


457. Slate Exercises. 


Add: 
16. 94 17. 273 18. 33,5; 19. 634 20. 871 
8} of 675 af 53g 
268 1003 924 or 9,1, 
21. 46,5, 22.36 23. 2751 24. 932 25.. 238 
52 743 54,3, 64 654 
1 91, 274 742 234 
2074 812 64 87 94 


458. Oral Problems. 


1. I spent 4 of a dollar fora ball and +4 of a dollar for a bat. 
What part of a dollar did I spend for both? 


2. += how many fourths? 


3. What will be the cost of a pen-knife at 3 of a dollar, and 
a book at 4 of a dollar? 


4. Write 4% with smaller numerator and denominator. 


5. I need 4 of a yard of ribbon for one hat and 4 of a yard 
for another. How much ribbon must I buy? 


6. Write a fraction equal to 5% with the smallest numbers 
youcan. (This is called reducing a fraction to lowest terms.) 


7. Sold 2 of a pound of tea to one customer and { to another. 
How much was sold to both? 


8. What quantity of oats must I buy to give $ of a peck to 
one horse and 3 to another ? 


9. If I sell 3 of a dozen of oranges to one person and } of a 
dozen to another person, what part of a dozen do I sell? 


10. What part of an hour is 45 minutes ? 


MIXED NUMBERS. 203 


11. 2 of an hour is how many minutes? 


12. I spent 4 of an hour reading and ,8, of an hour writing. 
What part of an hour did I spend at both? 


13. A boy is carrying 64 lb. of flour, and 62 lb. of ham. What 
is the weight of his load ? 


14. Reduce 43 to lowest terms. 
15. 18 hours is what part of a day? 


16. Reduce 4$ to lowest terms. 


ADDITION OF MIXED NUMBERS. 


459. Add 124, 62, 84, 158, 4. 


In the fractions 4, 2, }, 3, 3, the numbers above the line, 1, 2, 1, 5, 3, are 
called nwmerators; the numbers below the line, 2, 3, 4, 6, 8, are called denomt- 
nators. 

To add fractions, they must have the same denominator. An inspection 
of the denominators, 2, 3, 4, 6, 8, shows that 48 or 24 will contain each 
without remainder. 24, which is the smallest number that will contain 
all, is called the least common denominator. 


Oo 
121 | 12 
62 | 16 
81| 6 
158 | 20 

a| 9 


Instead of writing the common denominator 24, with each fraction, we 
may place it above, and write only the new numerators. 4=332, =}, 
1 = 6, etc. Write 12,16,6,20,9. The sum of these fractions is $3 = 2}$ = 23. 
5 is placed under the fractions to be added, 2 is carried to the whole numbers, 


making 43. 


Nors. — The fractional parts of answers should be reduced to lowest terms. 


204 ARITHMETIC. 


460. Slate Exercises. 


O> 
Oe 
oA 


19. 9921 + 888.4 778 
20. 1058+ 322 + 472 


Add: 

vast 2. 734 3. 932 4, 112 5. 181 
634 8} 24 32 72 
1% 393%5 4s 2055; 9.8, 
8 yo 16} a 54 4 

6. 125 + 355 + 27$ + 8} 

Te dont, SIERO 28 site ee 

Bib 7BR. cb OSE iat aioe 

9. Ste + 388F + 233 + 1 

10. 1003 + 754 + 9% + 494 

11. 332 + 178 + 245+ 69); 

12 Oly Oh Sea ee 

13. 103 + 842 + 253 + 98 

14,9188 0-2 3018s Sheen eons 

15. 4444 +5188 + 37, + 952 

is. 93%) 127aN- 18h pods 

17. Sie 1 64S) 1268 Oe 

18. 24 Ube aS ae eee 

-+- 
+ 


CO 
ol 


MULTIPLES AND FACTORS. 


461. A number that contains another number an exact num- 
ber of times is a multiple of that number. 

24 is a multiple of 12; 36, 48, etc., are also multiples of 12. 

30 is a multiple of 2, 3, 5, 6, 10, 15. 


MIXED NUMBERS. 205 


462. Oral Exercises. 
1. 95 is a multiple of what two numbers? 
2. Give two factors of 51. 
3. What number is a multiple of both 8 and 6? 


4. Mention another number that is a multiple of both 8 
and 6, 


5. Find the smallest number that can be exactly divided by 
8 and 12. 

6. Give two factors of 91. 

7. 571s a multiple of what two numbers ? 

8. What is the smallest number that can be exactly divided 
by 4, 6, and 8. 


463. The smallest number that is a multiple of two or more 
numbers is called the least common multiple of such numbers. 

9. Find the least common multiple of 5, 10, 15. 

10. Find the least common multiple of 2, 7, 14. 

11. What part of a dollar is 70 cents? 

12. 3—=how many 40ths? 

13. Reduce 172 to thirds. 

14. Change 7 to a mixed number. 

15. Reduce 24 to lowest terms. 

16. What part of a dollar is 15 cents? 

17. Add 4 and 4. 

18. From } take 4. 

19. Reduce 3% to 60ths. 

20. Which is largest, 2, #, or 3? 

21. $o0f 84=—? 

22. From 4} take s4. 

23. From 4% take 3. 

24. From 13 take +. 

25. From 44 take 3%. 


206 ARITHMETIC. 


SUBTRACTION OF MIXED NUMBERS. 


464. Sight Exercises, 


Subtract : 
1. 1614 2. 4919 3. 3828 4. 18,8 5. 275%, 
1355 37438 2944 143, 16}; 
6. 28,. 7 ATE 8. 3611 9. 2518 10. 321% 
13% 295 18355 1944 187%, 


465. Slate Exercises, 
11. 35 12. 68h 13. 278 14. 553 15. 105,38; 


— 82 — 975 —174 — 253 — 81 
16. 1208 17. 892.7 18) 188) 18s) OUR 20s 
— 847 — 884 — 155, — 214 — 594 
21. 480 22. 375 23. 200 24. 873 25. 1,000 
— 974 — 881 — 904 — 7572 — 9985 
26s rev ou 27 et 1G 28. 999 29. 132 30. 23 
— 2503 — 768 — 1382 — 46,3, —184 


466. Oral Exercises. 


—1=? l}-}=? 10-1=? 10}-1=? 10h-1}=? 


467. From 1972 take 682. 
15 Reduce the fractions to a common denominator (15), as in 
1972 9 addition of fractions. 42 being greater than ;%, we find the 
682 | 10 difference between +2 and 1,%, or 24. Writing this difference, 
12914 | 14 1$, we add 1 to 68 and subtract. 
Loe. & 


MIXED NUMBERS. 207 


468. Subtract: 
31. ~ 8} 32. 231 33. 344 34. 161 35. 211 


5} 58 272 81 ga 
Bomcey ennsT Soh, 88. 77) 884) D014) 40. 998 
293 14 594 244 881 
41. 1008, 42. 25,8, 43. 93,2 44. 10199 45. 122 
763 51 O44 983. 48 
oe eee 48. 188%. 49. 161. 50... B7E 
loz5 3} 277% 348, 294 
pimoeee 52.) 642 S08, 125 te) BA ATE! BB ys Tod 
495 181 1008 88 504 
Bos ale 957) 63, 58. Sy, 69. 252. 60. 1025, 
273° 445 1} 174 864 


469. Slate Problems. 


1. From a piece of cloth containing 174 yards, 53 yards and 
42 yards were sold. How many yards were left? 


2. A boarding school uses 3 quarts of milk a day for 7 pupils. 
If there are 77 pupils in the school, how many gallons of milk 
will be needed per day? Per week? 


3. A man pays $140.40 for 3 pieces of cloth. What is the 
length of each piece, if the cloth costs $1.80 per yard? 

4. I buy 12% pounds coffee at 28 cents per pound, and twice 
as many pounds at 24 cents per pound. If I give the store- 
keeper $10, how much change will I receive? 

5. A merchant pays $30 for 65 vases. He sells 17 of them 


at 40 cents each, 23 at 60 cents each, and receives 48 cents each 
for the others. What is his profit? . 


208 ARITHMETIC, 


6. Mrs. Jones buys 4 dozen chairs, a bureau for $17.40, and 
a mirror for $18.00. She pays for all $38.80. What do the 
chairs cost apiece ? 


7. In selling 40 yards of velvet that cost me $1.40 per 
yard, I gained $24.00. What did I charge per yard? 


8. How much heavier is a cheese weighing 403 pounds than 
one which weighs 26% pounds? 


9. A farmer sold 364 dozen eggs to one storekeeper, 52 
dozen to another, 17% dozen to a third, 83 dozen to a fourth, and 
11,4 dozen to a fifth. How much did he receive for them at 12 
cents per dozen? 


10. A butcher going to market with $174.40, bought 15 sheep 
at $8 each. If he had had $25.60 more, he could have bought 
4 hogs also. What were the hogs worth apiece ? 


11. A scholar having to multiply a number by 30, mistook 
the 8 for a 5, and his answer was 600. What is the correct 
answer ? 


12. Forty hats were bought for $104. At what price apiece 
should they be sold to make 40 cents on each hat? At what 
price per dozen ? 


13. A farmer had 7 bushels of potatoes. He used 2 bushels 
and 3 pecks for seed. What would the remainder be worth at 
20¢ per peck ? 


14. A butcher buys an ox weighing 1,200 pounds alive, at 
6 cents per lb. When killed and dressed, its weight is 2 of the 
live weight. What is the butcher’s profit, if he sells the meat at 
an average of 15 cents per lb.? 


15. The weight of a tub of butter, including the weight of the 
tub, is 484 pounds. The tub weighs 94 lb. What is the butter 
worth at 24 cents per pound ? 


MIXED NUMBERS. 209 


16. One boy had 15 marbles, another had 19, a third had 17, 
a fourth had 18. What was the average number of marbles for 
each boy? 


17. A teacher divided 200 foreign postage stamps among the 
eight boys of his class. He gave one-fourth of them to the first 
boy, one-fifth of the remainder to the second boy, and then 
divided the rest equally among the other six boys.. How many 
did each receive? 


18. A merchant sold 17% yards of muslin, 144 yards of silk, 
and as many yards of calico as of the other two together. How 
many yards did he sell in all? 


19. A dealer mixed 23 pounds of black tea costing 32 cents 
per pound with 14 pounds of green tea costing 40 cents per 
pound. How much per pound does the mixed tea cost him? 


20. A man can do 4 of a certain piece of work in a day; 
another man can do } of the same work in a day. What part 
of the work can both together do in a day? How long would it 
take both together to finish the work ? 


470. Slate Exercises. 


Find answers: 


ha atte 2. 124 3. 282 4. A474 5. 28 
x 7h x 192 x 8 x 12 x 48 
ibe Aes 7. 848 St 975 69 102) Sl 
x 1062 fxd x 3h x 44 x 154 
471. Divide: 
11. 9383+3 15. 808 +5 19. 8252211 
12. 564+4 16. 126126 20. 1,72812 +12 
13. 985 +7 17. 450% + 9 21. 2,46012 + 10 


14. 282-2 18. 3608 =8 22. 3,92613 + 13 


210 ARITHMETIC. 


NOTATION AND NUMERATION. 


472. The largest number that can be written with six figures 
is 999,999. 

1,000,000 is called one million. 

Write in figures two million. Three million. Four million. 
Six million. Eight million. Ten million. 


473. Read the following: 


1. 1,234,567 6. 11,034,065 11. 30,100,021 
2. 8,000,560 7. 14,602,500 12. 35,000,600 
3. 5,009,008 8. 17,886,925 13. 401,023,160 
4. 7,090,070 9. 20,007,316 14. 760,030,020 
5. 9,843,000 10. 25,000,005 15. 980,750,000 


474. Write in figures: 
1. Seventy-eight million, one hundred eight thousand, ninety- 
SIX. 
2. Three million, eight. 
3. Fourteen million, seven thousand, five. — 


4. Nine hundred. eighty-seven thousand, six hundred fifty- 


5. Twenty million, thirty thousand, forty. 

6. Three hundred seven million, nine hundred four thousand, 
SIX. 

7. Nine hundred ninety-nine million, nine hundred ninety- 
nine thousand, nine hundred ninety-nine. 


8. Four hundred seventy-six million, three hundred thou- 
sand. 


9. Thirty-four thousand, eighteen. 
10. Sixty-four million, thirty-two thousand, sixteen. 
Add the foregoing. 


REVIEW. 


475. Review. Slate Exercises, 


Read the following numbers. 


1. 27,088,549 
116,908,070 
3,006,005 
20,080,070 
1,647,893 
206,045 

73,000 

180,059 
2,316,045 
54,006,000 


2. 508,900,007 


68,487,291 
4,629,880 
25,936,097 
134,870,603 
59,009,300 
7,000,004 
686,909 
50,308 
9,999 


476. Add down and across: 


Lt 

20 + 

300 + 
4,000 + 
50,000 ++ 
600,000 + 


24 
30 + 
400 + 
3,000 + 2, 
40,000 + 30, 
500,000-+ 400, 


3+ 
40 + 
500 + 
000 + 
000 + 
000 + 


3. 


Add each column. 


248 576,908 
36,200,570 
5,987,600 
380,070 
68,000 
593,056 
2,384,672 
59,876,004 
123,321,123 
88,888,888 


4=? 

50 =? 

600 =? 
1,000 ==? 
20,000 = ? 
300,000 =? 


7,000,000 + 8,000,000 -+ 6,000,000+ 5,000,000 = ? 
30,000,000 + 40,000,000 -+ 50,000,000 + 60,000,000 = ? 


fmol -t, 


foe 


4. 


(nd tts 


212 


477. Multiply across. 
10x 2=? 
10x 5=? 


10x ?=? 


pipe gy agg 
TAs ordinal LOK —=9¢ 


Dia Wo CAM sr ¢ 


123xX 3=? 
123 x 20=? 
123 x 100 =? 


123x ?=? 


478. Add: 


1. $184,635.873 
25,904.63 
8,756.95 
889.57 

4.326.981 
58,030.05 
209,508.67 
37,654.88 

9,876,544 
68,250.89 
6,505.83 


ARITHMETIC. 


2. $263,005.95 


38,462.77 
159,076.50 
50,318.92 
36,485.73 
9,860.44 
76,035.00 
8,900.56 
4,056.02 
26,465.54 
7,826.98 


Add multipliers and products. 


A a 
loo a 
IZWxX Ghee 
50 x Wai? 
50x 26==2 
50 xe 
560 x oi 4a 
560 <x 50-27 
560 x 600 = ? 
CX 

8. $5,503.00 

45 837.24 

24,328.49 

567,849.60 

20,486.04 

54 200.30 

2,578.64 

328.99 

3,468.36 . 
87,243.25 


28,376.42 


i 


MIXED NUMBERS. 


MULTIPLICATION OF MIXED NUMBERS. 
483. Sight Exercises. 


Give answers: 


2of 9 


2. Sof 7 
3. 
4. 2o0f17 


2 of 11 


9x2 
7x % 


. 11x? 
. 17x % 


484. Slate Exercises, 


25. 
26. 
27. 
28. 
29. 
30. 
31. 


9x 12 
7X 23 


32. 
33. 


(Er fe 
Se 
19. 
20. 


19 x 8% 
23 x 9b 


Oo tauw 


- 4 


3 of 15 
# of 20 
4 of 18 


io) 

ma 
1 
a 


15 Xx 
20 X 
13 xX 
14 x 


Oe col alco alco 


11 x 38 
17 x 43 
15 x 58 
20 x 68 
13 x 7% 


46. 887 
x 4003 


47. 1,986 
x 8,0754 


48. 2,340 
x 1,6073 


34. 17 x 103; 
35. 122x 9 


5836. 133 17 


37. 1438 x ll 
38. 152 x 17 


49. 698 
x 858 


50. 1032 
x 275 


51. 41033 
preen 


9. 
10. 
Ag I 
uF 


21. 
22. 
23. 
24. 


39. 
40. 
41. 
42. 
43. 
44. 
45. 


52. 


53. 


54. 


z°5 of 19 
34, of 30 
335 of 17 
34, of 10 


19 x 3% 
23 X 
17 x 
10 x +5 


163 xX 
173 x 
18% x 
19.2, X 
2054 X 
21,5; x 
22715 X 


81,5094 


x19 


23082 


x 1,560 


6,0898 


x 3,520 


213 


214 ARITHMETIC. 


55. 2,3403 57. 1,8572 59. 3,579, 
x 8,004 x 2,060 x 4,300 

56. 4,060 58. 3,050 60. 2,469 
x 2,0502 x 2,0602 x 4,987, 


DIVISION OF MIXED NUMBERS. 
488. Oral Exercises, 


How many times is 4 of a dollar contained in $1? How many 
times is + of a pint contained in 1 pint? + of a gallon in 1 
gallon ? 

How many times is $ of a dollar contained in $2? In $3? 
In $5? 

How many times is 4 of a dollar contained in $1.50? In $2.50? 
In $3.50? In $4.50? 

How many times is 1 half contained in 3 halves? In 5 halves? 
In 7 halves? In 9 halves? 


B+h=?  §4$=?  F4d=? 0 g+d=? 

How many times is 3 contained in 8? In}? In 48? In 221? 

Divide li by 14. 44 by 14. Th by 14. 10) by Li. 

Divide 3by 14. 6by 14. Oby ld. 12by 14. 15 by 1k. 

Divide 5 by 14. 64 by 14. 10 by 14. lli by 1}. 15 by 14. 

Divide $ by 3. $by % 12 by % ld by % 2tbys 38 by 3. 
3% by 2. 


489. Slate Exercises. 
Divide 250 by 123. 
123 = 25 halves. 


250 = 500 halves. 
500 halves + 25 halves = 500 + 25 = 20, Ans. 


MIXED NUMBERS. 


Divide 1,3874 by 183. 


183 = 75 fourths. 


Change 1,387 to fourths by multiplying by 4. 
1,3874 x 4 = 5,550; that is, 1,3874 = 5,550 fourths. 
75 fourths is contained in 5,550 fourths 74 times. 


215 


Ans. 74 
75) 5550 
3800 

00 


490. Divide. Prove the correctness of the answer by multi- 


plying the quotient by the divisor. 


491. 


492. 


Le 


2 
3. 
4 
5 


75 +124 


. 150+124 


75+ 64 


. 150+ 64 


62-+ 152 


624 + 124 


eievereabs 


814+ 64 


. 19388- 61 


77h + 154 


. 192, +174 


978+ 53 


. 1988+ 738 
. 1662+ 832 
. 1881 + 162 


. 60 +38 
. 60 +74 
. 603+3 
. 604+ 2 
. 624+ 64 


6 
7. 
8 
9 


10. 


- 105+174 
69+ 52 
93+ 72 

- 100+ 334 
150 + 162 

60+ 4 
60 +14 
60-+ 4 
60-14 
60+ 4 
60 + 12 
60+ 2 
60 + 21 
60+ 2% 
60 + 34 
871 + 64 
153 + 18 
244 + 13 
871 = 8% 
494 + 44 


216 ARITHMETIC. 
493. 
41. 75 + 34 48. 183 +2, 
42. 125 + 34 49. 334+ 2,1, 
43. Sd3i+ 3} 50. 294+ 274, 
44. 1908+ 34 51. 321+ 31 
45. 100 +113 52. 502+ 44 
46. 100 + 14 . 53. 512+ 52 
47. 121+ 2,4 54. 613+ 64 


FEDERAL MONEY. 
494. Slate Exercises. 
Add the following without placing the amounts in columns. 
. $8.34, $40.39, $638.27, $594.38, $1.97. 
$0.03, $8.05, $600.00, $38.72, $198.52, $0.63. 
$432.84, $96.25, $3.64, $782.46, $800.06, $6.50. 
. $8.60, $40.05, $91.86, $350.48, $84.00, $287.63. 
$98.27, $0.60, $600.39, $8.09, $37.88, $503.07: 
$202.97, $42.23, $453.60, $7.18, $63.54, $0.37. 
$8.48, $0.54, $2.57, $85.18, $425.31, $8.27. 
$486.54, $84.62, $1.96, $8.18, $35.84, $236.49. 
$83.61, $523.00, $23.04, $0.86, $35.82, $584.60. 
$34.80, $98.54, $200.41, $324.86, $50.14, $8.75. 


OM RST PF wo Yb 


— 


The foregoing examples may be added directly from this 
book or from the blackboard, the pupils writing on their slates 
nothing but the answers. 


FEDERAL MONEY. 2A 


495. Subtract the following without re-arranging them. Find 
the sum of the minuends, the sum of the subtrahends, and the 
sum of the remainders. 


11. $1,000.00— $876.49 = 
12. $54937— $99.89= 
13. $345.93— $76.04= 
14. $1,786.08 — $1,097.19 = 
15. $345.00— $187.23—= 
16. $3,545.37 — $966.38 = 
17. $8246—  $7.59== 
18. $5,074.02 — $4,987.63 = 
19) $77.84— $9.88 = 


20. $4,680.35 — $4,679.46 = 
Sie ? —_ Te = ? 


496. Multiplication. 
Find the cost of : 


1. 


CO APT Pw NS 


- 
= 


197 barrels of flour, at $5.66 per barrel. 

486 bushels of wheat, at $1.04 per bushel. 
237 tons of plaster, at $6.72 per ton. 

809 tons of hay, at $11.45 per ton. 

74. car-loads of bran, at $20,622 per load. 
208 sheep, at $4.65 per head. 

673 barrels of mackerel, at $10.60 per barrel. 
984 bushels of onions, at $1.09 per bushel. 

99 pounds of butter, at 24 cents per pound. 
208 pounds of coffee, at 28 cents per pound. 


218 ARITHMETIC. 


497. What will be the cost of 157 pounds of sugar, at 5% per 


pound. 

157 At 5% per pound 157 pounds will cost 157 times 5%. In prac- 
5 tice, however, we multiply 157 by the smaller number 5. 

785 Ans., $7.85. 


11. 1,376 yd. of muslin, at 63f. 

12. 2,084 bu. of corn, at 471f. 

13. 1,864 lb. of beef, at BLY. 

14. 988 lb. of turkeys, at 181f. 

15. 296 bu. of potatoes, at 471. 

16. 1,272 lb. of dried apples, at 23¢. 
17. 488 lb. of lard, at 103¢. 

18. 2,240 lb. of sugar, at 48¢. 

19. 5,176 lb. of wool, at 308¢. 

20. 4,892 bu. of wheat, at 993¢. 


FRACTIONAL PARTS OF A DOLLAR. 


499. What will be the cost of 16 base-balls at 25 cents each? 


Multiplying 25 cents by 16, we get the answer $4.00. A shorter way is 
14 multiply 4 of a dollar by 16 which gives 16 quarter-dollars, or 4 dollars. 


500. Oral Exercises. 
At 25 cents per lb., yd., doz., etc., what will be paid for : 


1. 32 base-balls? 7. 387 doz. lemons? 

2. 52 |b. of coffee? 8. 25 bu. of tomatoes? 

3. 48 straw hats? 9. 41 panes of glass? 

4. 84 yd. of ribbon? 10. 33 packages of candy? 
5. 36 second readers? 11. 49 Roman candles? 
6.. 56 vases? 12. 60 bars of soap? 


‘FEDERAL MONEY. 219 


501. At 50 cents, give the cost of: 


13. 46 lb. of tea. 19. 76 grammars. 

14. 28 pairs of scissors. 20. 57 boxes of pens. 
15. 38 penknives. 21. 49 picture books. 
16. ‘84 third readers. 22. 83 dolls. 

17. 44 lb. of candy. 23. 27 games. 

18. 32 caps. 24. 75 feather dusters. 


25. What part of a dollar is 75 cents? 


26. If I pay 3 quarter-dollars each for sleds, how many quar- 
ter-dollars will I pay for a dozen sleds? How many dollars? 


502. Give the cost of the following at 75 cents per yd., etc. 


27. 16 pairs of skates. 33. 21 bu. of rye. 

28. 11 yd. of silk. 34. 22 shad. 

29. 24 bu. of peaches. 35. 13 Ib. of tea. 

30. 15 gal. of syrup. 36. 30 gal. of ice-cream. 
31. 9 pairs of gloves. 37. 7 mats. 

32. 18 hats. 38. 17 concert tickets. 


39. How many cents in one-eighth of a dollar? 


40. At one-eighth of a dollar each, what will be the cost of 
24 bars soap? 


503. Give the cost of the following at 124 cents per lb., etc. 


41. 16 lb. of meat. 45. 80 jars of jelly. 

42. 48 doz. eggs. 46. 96 cans of condensed milk. 
43. 72 qt. of plums. 47. 104 yd. of sheeting. 

44. 64 gal. of oil. 48. 88 lb. of currants. 


49. How many cents in one-third of a dollar ? 


50. At one-third of a dollar each, what will be the cost of a 
dozen bottles of cologne ? 


220 


ARITHMETIC. 


504. Give the cost, at 334 cents per yd., lb., etc., of: 


51. 36 yd. of ribbon. 
52. 63 pairs of cuffs. 
53. 48 lb. of butter. 


505. Parts of a Dollar. 


64 cents = |; of $1 
81 cents = 7, of $1 
123 cents= tof $l 
162 cents= 4 of $1 
25 cents= iof$l 
334 cents= 4of $1 


506. Oral Exercises. 


Give the cost of 72 articles at: 


57. 123 cents each. 
58. 374 cents each. 
59. 624 cents each. 
60. 874 cents each. 
61. 334 cents each. 


507. Multiply : 


67. 64centsx 16 
68. S8icentsx 24 
69. 123 cents x 88 
70. 162 cents x 954 
71. 25 cents x 240 
72. 334 cents X 66 
73. 3724 cents x 48 


54. 27 bu. of oats. 
55. 54 pk. of walnuts. 
56. 72 doz. oranges. 


374 cents = 3 of $1 
50 cents =4 of $1 
621 cents = 3 of $1 
662 cents = 2 of $1 
75 cents = 2 of $1 
874 cents = 7 of $1 


62. 662 cents each. 
63. 84 cents each. 
64. 25 cents each. 
65. 50 cents each. 
66. 75 cents each. 


74. 50 cents x 186 
75. 621 cents X 32 
76. 662 cents xX 33 
77. 75 cents X 128 
78. 874 cents x 88 
79. 162 cents x 246 
80. 334 cents x 156 


FEDERAL MONEY. 221 


81. 374 cents x 80 
82. 75 cents x 160 
83. 874i cents x 24 
84. 623 cents x 64 
85. 64 cents x 160 
86. 124 cents x 168 


508. Find the cost of: 
93. 86 neckties, at 50 cents each. 


509. Blackboard Exercises. 


87. $1.3831 x 24 
88. $1.12ix 16 
89. $2.25 x 12 
90. $3.75 x 12 
91. $4371 x 8 
92. $5.162x 6 


94. Six dozen handkerchiefs, at 25 cents apiece. 
95. 32 yards of silk, at $1.124 per yard. 

96. 64 arithmetics, at 75 cents each. 

97. 84 geographies, at $1.25 each. 

98. 96 lb. of tea, at 75 cents per pound. 


99. 84 pairs of gloves, at $1.50 per pair. 
100. 72 yards of cloth, at $2.123 per yard. 


Write answers. 


Do not write quantities or prices on the slate. 


A 


687 pounds, at 4. 
976 yards, at 6f. 
2.504 dolls, at 25¢. 
352 yards, at 121f. 
1,728 hats, at 50¢. 
933 yards, at 331. 
938 coats, at $7. 


8. 

9. 
10. 
11. 
12. 
135 
14. 


695 pounds, at 20¥. 
248 pounds, at 75¢. 
186 pounds, at 662. 
12 bushels, at $1.43. 
9 tons, at $22.75. 

8 barrels, at $16.374. 
11 sheep, at $7.47. 


POY, ARITHMETIC. 


15. 16 gallons, at $3.624. 18. 96 pounds, at $1.25. 
16. 13sacksofsalt,at $1.11. 19. 120 gallons, at $2.33. 
17. 124 bushels, at $1.50. 20. 64 sacks, at $1.121. 


510. Oral Exercises. 


At 50 cents each, how many penknives can be bought for $1? 
For $2? For $38? For $10? For $20? 

At 25 cents each, how many readers can be bought for $1? 
For $2? For $8? For $10? For $20? 

At 124 cents per yard, how many yards can be bought for 
pl? For$2? For$3? For$10? For $20? 

At 334 cents per pound, how many pounds can be bought for 
$1? For$2? For $3? For $10? For $20? 


511. Oral. 
At 25 cents each: 

1. How many base-balls can be bought for $9 ? 
Straw hats, for $12? 
Roman candles, for $18? 
Readers, for $15? 
Vases, for $21? 
Bars of soap, for $31? 
Packages of candy, for $41? 
Yards of ribbon, for $5.75 ? 
Bushels of tomatoes, for $10.50? 
Pounds of coffee, for $12.75? 


CO AST P OW 


= 


512. At 50 cents: 
11. Pounds of tea, for $43? 
12. Penknives, for $20.50? 
13. Pounds of candy, for $94? 
14. Third readers, for $17.50? 


15. 
16. 
17 
18. 
19. 
20. 


FEDERAL MONEY. 


Caps, for $21? 

Grammars, for $37? 

Boxes of pens, for $72? 
Dolls, for $64? 

Pairs of scissors, for $19? 
Feather dusters, for $26.50? 


513. At 121 cents: 


514. At 33 


21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 


Gallons of oil, for $8? 

Dozen of eggs, for $11? 
Pounds of meat, for $21? 
Quarts of plums, for $14? 
Jars of jelly, for $2? 

Yards of sheeting, for $14? 
Cans of milk, for $21? 
Pounds of currants, for $ 3.124 
Whisk brooms, for $4.372? 
Collars for $5.624? 


1 
= cents: 


31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 


Yards of ribbon, for $6? 
Pairs of cuffs, for $12? 
Pounds of butter, for $18? 
Bushels of oats, for $32? 
Pecks of walnuts, for $14? 
Dozen of oranges, for $12? 
Straw hats, for $2.332? 
Dolls, for $3.662? 
Penknives, for $ 4.854? 
Pounds of candy, for $5.662? 


223 


224 ARITHMETIC. 
515. At 162 cents: 
41. Collars, for $4? 46. 
42. Pounds, for $21? 47. 
43. Yards, for $4? 48. 
44. Ounces, for $4? 49. 
45. Packages, for 662 cents? 50. 
516. Blackboard Exercises, 
Divide at sight : 
51. $24.50 by 50 cents. 56. 
52. $12.25 by 25 cents. 57. 
53. $26 by 334 cents. 58. 
54. $14.50 by 124 cents. 59. 
55. $17 by 162 cents. 60. 


Quarts, for $1.162? 
Gallons, for $1.50? 
Pecks, for $22? 
Feet, for $3.33}? 
Yards, for $4.662 ? 


$18.75 by 25 cents. 
$11.874 by 124 cents. 
$13.33 by 334 cents. 
$37.50 by 50 cents. 
$13.334 by 162 cents. 


DIVISION OF FEDERAL MONEY. 


517. Oral Exercises. 
How often is 1 quart contained in 1 gallon? 1 pt. in 1 qt.? 


2 qt. in 1 gal.? 


1 inch in 1 foot?’ ° 2inv in) 1ttit Sinan ite 


4-in.in. 1: ft.? Goi, in yt? Grin in) 2 fhe ie he aes 


1 ounce in 1 pound? 


in wohalt7 >.) third inal 
How often is 1 cent contained in $1? 2 cents in a dollar? 
4 cents in 2 dollars? 25 cents in 25 dollars? 


518. Give answers at sight: 


PO wD 


$4 + 10¢ 
$5+ 5¢ 
Pee 12e 4 
. $86+ 69 


leozsin 2b? 4-02, anv lb es laouren 


5. $638+ 3f 
6: B74 25¢ 
7. $20+3381¢ 
8. $386+ 3¢ 


FEDERAL MONEY. 225 


9. $40+50¢ 15. $16+16¢ 
10. $9-+10¢ 16. $16 + 162¢ 
11. $1+i3/ 17. $16+ 331¢ 
12. $38+$1 18. $16-- 25¢ 
13. $84-+ 50¢ 19. $16 + 50¢ 
14. $1+ 163¢ 20. $12 + 20¢ 


519. At 36 cents each, how many spellers can be bought 
for $27? 
75 


36)2700 $27 = 2,700 cents. Since 1 speller costs 36 cents, for 2,700 
180 cents we can buy 239° spellers. Ans. 75 spellers. 


0 


520. Slate Problems. 


1. At $2.75 per day, how long will it take a man to earn 
$110? 
11,000 + 275 


2. How many yards of muslin, at 12 cents per yard, can be 
bought for $126 ? 


3. A farmer spent $140 for sheep at $5.60 each. How many 
did he buy? 


4. A grocer pays $74.50 for tea at 4 of a dollar per pound. 
What is the weight of the tea? 


5. When rye is worth 87 cents per bushel, how many bushels 
can be purchased for $261 ? 


6. At 124 cents per pound, how many pounds of meat will 
cost $175.25 ? 


7. If '75 spellers cost $27, what is the price of 1 speller? 


8. A woman paid $24 for 36 yards of dress goods. What 
did she pay per yard? 


226 ARITHMETIC. 
9. At 6 for a dollar, how many rabbits can be bought for 
$87? 


10. The cost of 13 houses was $36,887.50. What was the 
price of each ? 


521. Sight Exercises. Approximate Answers, 


1. What will be the cost of 389% lb. butter at 20¢ per lb.? 
(Nearly 40 lb. at 20%. The cost is nearly what ?) 


2. Aman has 4,200 pounds of flour which he wishes to put 
into barrels containing 196 lb. each. About how many barrels 
will he need ? 


(Each barrel contains nearly how many pounds ?) 


3. A merchant bought a hogshead of molasses, containing 472 
gallons, at 50 cents per gallon. About how much did it cost? 


4. How many lots at $1,975 each can be bought for $12,000? 


5. Sold 3 pieces of cloth, 33 yd. to the piece, at $1.95 per yd. 
Give the approximate amount of the bill. 


6. 2813 + 3748 = nearly what? 
7. 175} + 24,9 = nearly what? 
8. 18% x 92 = nearly what? 

9. 87, — 4918 = nearly what ? 


10. 43 x 48 x 4,9, = nearly what? 


525. Learn the following tables: 


TIME. 


€0 seconds (sec.) 1 minute (min.) 
60 minutes 1 hour (hr.) 

24 hours 1 day (da.) 

7 days 1 week (wk.) 


DENOMINATE NUMBERS. 27 


Dry MEASURE. 
2 pints (pt.) 1 quart (qt.) 
8 quarts 1 peck (pk.) 
4 pecks 1 bushel (bu.) 


AVoIRDUPOIS WEIGHT. 


16 ounces (oz.) 1 pound (Ib.) 
2000 pounds 1 ton (T.) 


The hundredweight (100 pounds) is written cwt. 


Liquip MEAsuRE. 
2 pints (pt.) 1 quart (qt.) 
4 quarts 1 gallon (gal.) 
A gill (gi.) is equal to one-fourth of a pint, It is very rarely used. 


DENOMINATE NUMBERS. 
526. Slate Exercises, 


1. How many hours in 74 days? 
2. How many hours in 7 days 12 hours? 
3. How many seconds in 2 hours? 


4. A man buys 12 bu. and 3 pk. apples @ $1 per bu. What 
is the cost? 


5. What will be the cost of 8 pk. 7 qt. chestnuts @ 8¢ 
per qt. ? 

6. How many pints are there in 5 gallons of ice cream ? 

7. How many half-pints are there in 10 gallons of ice cream ? 


8. How many 4-ounce packages can be made from 5 pounds 
of pepper? Y 


9. A boy pays $1.50 for 1 gallon and 2 quarts of ice cream. 
What is the price per quart? 


228 


10. 
people 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
16 yd. 
29. 


ARITHMETIC, 


How many gallons of lemonade will be needed to give 96 


+ pt. each? 


How many seconds in 5 hours? 

How many minutes in 1 week ? 

Change 13 hours and 20 minutes to minutes. 
Change 15 bu. and 4 pk. to pecks. To quarts. 
How many ounces in 47 lb. 5 oz.? 

How many pounds and ounces in 237 ounces? 
Change 1,494 minutes to hours and minutes. 
Find the number of hours in 6 weeks. 

How many hours are there in January ? 

How many inches are there in 2 yd. 2 ft. and 2 in.? 
How many ounces in 4 tons? 

Reduce 5 cwt. and 80 lb. to oz. 

How many pounds in 3 of a ton? 

In ? of a ton how many cwt.? 

What will 400 lb. of coal cost at $5 per T.? 
What fraction of a ton is 1,500 lb. ? 

How many days and hours in 2 of a week? 


Find the number of yards in 3 pieces of cloth averaging 
2 ft. each. 


When coal is $5.50 per ton, how much will I have to 


pay for 3,000 pounds ? 


30. 


A store-keeper charged 70 cents for a roll of butter weigh- 


ing 2lb.30z. What was the price per ounce? What was the 
price per pound ? 


31. 
32. 


Change 60 lb. to the fraction of a hundredweight. 


How many pints are there in a barrel of oil that contains 


434 gallons? 


REVIEW. 


SPECIAL DRILLS. 


528. Give sums: 


56 + 25 
47 +47 
22 + 68 
39 + 31 


529. Give remainders: 


81 — 56 
94 — 47 
60 — 28 
72 — 39 


530. Give products : 


310 x 9 
420 x 4 
630 x 3 
740 x 2 


531. Give quotients : 


196 + 4 
190 +5 
192+6 
196 + 7 


532. Give answers: 


O14 1b 
+1 
+1 
+1 


21+ 32+ 48 750 + 190 
30 + 20+ 18 390 + 120 
40+ 18+ 26 480 + 150 
24 + 31+ 30 620 + 180 
750 — 190 750 — 560 
510 — 120 510 — 890 
630 — 150 630 — 480 
820 — 160 820 — 660 
aoe 207 
65 x 3 24x 8 
49 x 4 APA oe 
AYE pada Leap a 
196 + 49 450 + 25 
190 + 38 375 + 25 
192 + 32 225 +- 25 
196 + 28 350 + 25 
1i— 2 Zof 66 
21 — 14 84x 2 
bt — 24 4 of 100 
41 — 31 186x 2 


229 


225 + 54 
315 + 21 
437 + 60 
540 + 55 


— 

Comb 

la 
ey 


On 


ise) 
oye ess 
| 


o> cto wmloo bole 


230 ARITHMETIC. 


533. Oral Problems. 


1. Paid 59¢ for muslin and 25¢ for trimming. How much 
was paid in all? 


2. A boy had 75%. How much had he after spending 25¢ 
for a knife and 15¢ for a ball? 


3. If 8 lb. of raisins cost $1.04, what is the price per pound? 
4. At $1.89 per yard of silk, what will be the cost of 1 foot? 


5. If 382 lb. of flour cost 96 cents, how many pounds can be 
bought for 60 cents? 


6. One girl has 16 cents, another has 24 cents, a third has 8 
cents. How many dolls at 16 cents each can be bought with 
their money ? 


7. What will be the weight of 3 bushels corn, weighing 56 
pounds per bushel ? 


8. How many ounces in 9 pounds? 


9. How many pounds in 8 packages, each weighing 16 


ounces ? 


10. Find the cost of 3 lb. and 2 oz. of butter at 32 cents 
per lb. 


11. Bought 4 pounds of 6-cent sugar and a pound of butter at 
86 cents. How much change from $1? 


12. Four boys have 144 marbles among them. If they were 
equally divided, how many would each have? 


13. A man earns $100 per month, and spends $76. How 
much does he save? 


14. If a man saves $32 per month, how many months will 
it take him to save $960? 


15. Paid $27.90 for 9 jackets. What did they cost apiece? 


REVIEW. Zoos: 


16. Mr. B's farm contains 520 acres. How many acres will 
he have left after selling 180 acres? 


17. William’s kite string is 435 yards long, John’s is 62 yards 
longer. What is the length of John’s string? 


18. A farmer raised 168 bushels of grain. He sold 4 of it. 
How many bushels did he sell ? 


19. 64 yards of ribbon are cut into pieces a quarter of a yard 
long. How many pieces are there? 


20. Ifit takes 18% yards of cloth to make 3 suits, how many 
yards does it take for 1 suit? 


21. James has 150 marbles, Thomas has # as many. How 
many marbles have both? 


22. A newsdealer received $6.36 for papers sold at 3 cents 
each. How many papers did he sell? 


23. If it takes 44 days for one man to do a piece of work, how 
long will it take 2 men to do the same work? | 


24. A farm is divided into 4 fields, each containing 49 acres. 
How many acres are there in the farm? 


25. From a piece of cloth containing 104 yards, 53 yards are 
sold. How many yards are left? 


26. Find the cost of 28 lb. coffee at 25% per lb. 


27. How much does a farmer receive for 28 cows which he 
sells at $25 each? 


28. Find the number of hours in a week. 


29. How many pieces, each three-quarters of a yard long, can 
be cut from six yards of wire? 


30. 3,600 seconds are equal to how many minutes? 


31. If 25 yards of material are needed for a dress, how many 
yards will be required for 33 dresses? 


32. At 7 for a cent what will 98 marbles cost ? 


232 


534. Sight Exercises, 
Divide : 


CL Wate 


535. Give answers in whole numbers. 


20. 


960 + 240 
780 + 260 
960 + 480 
720 + 180 


1,170 + 130 


960 + 241 
779 + 260 
959 + 480 
720 +181 


1,030 + 1380 


536. Multiply: 


ol. 
32. 
33. 
34. 
35. 


537. Perform indicated operations : 
18 + (80 x 4) 
7+(2x 8)—4 
[((7 +2) x 8]—4 
Lead) 
(6xP)+4 

1 of 4 of 600 


1,200 x 6 
1,800 x 4 
2.500 x 3 
1,700 x 5 
1,400 x 7 


51. 
52. 
53. 
54. 
55. 
56. 


36. 
37. 
38. 
39. 
40. 


ARITHMETIC. 
6. 8400+ 2,100 Ain ie 
7. 8,600 + 4,300 ib} 
8. 8,800 + 2,200 13. 
9. 9,600 -- 3,200 14. 

10. 9,900 + 8,300 15. 


10,800 = 1,200 
10,400 + 1,300 
6,000 ~~ 1,500 
5,700 -- 1,900 
12,000 + 2,400 


(Omit remainders.) 


21. 8400+2,110 26. 10,800 +1,205 
22. 89500+4,3800 27. 10,300~+ 1,800 
23. 8800+2,199 28. 6,100+ 1,550 
24. 9599-3199 29. 5,700+ 1,899 
25. 10,000-+3,3380 30. 12,020+ 2,410 


1,800 x 9 
2,300 x 3 
8,200 x 2 
1,500 x 4 
1,200 x 8 


41. 
42. 
43. 
44. 
45. 


2,100 x 4 
1,400 x 6 
4,100 x 2 
1,600 x 5 
2,200 x 3 


46. 
47. 
48. 
49. 
50. 


1,400 x 8 
2,400 x 4 
15300 Sra 
1,200 x 9 
6,300 x 2 


57. 4 of (240+ 60) 
58. (7+2)x(8—4) 
59. 7+[2x(8—4)] 
60, ate 
61. 6X (4+4) 
62. $x 12x23. 


538. Place the following upon the board. The pupils 


write the answers and nothing else. 


582 
97 
617 
77 
299 
36 
458 
57 


63. 


64. 


65. 


66. 


67. 


68. 


69. 


70. 


209 
29 
374 
45 
582 
87 
292 
56 


539. Slate Exercises. 


Add: 
1. $3,947.25 
14,816.00 
956.83 
2,469.98 
95,783.04 
9,005.79 
6,598.86 
1.58 
39.99 
463.27 
85 


4. $18,477.09 
494.78 
1,489.07 
104.84 

91.03 
20,999.99 
7,583.94 
87.62 

3.47 

6,952.83 


REVIEW. 


233 


will 

(Art. 385.) 
907 524 Be Ol 
1. — 5. — OF nes 
. 93 f 76 a 85 
470 430 295 
2. — 6. — ._—_ 
67 u 86 oh 66 
260 310 876 
3. — ._— Poms 
ui 55 bd 54 : 95 
400 865 573 
74. — 8. — 82. — 
Tie keno mee ae 

Review, 

2. $14.92 3. $1,094.07 
3,120.50 789.14 
18.72 9,870.00 
79,841.24 4,009.89 
3,972.87 484.78 
104.99 9,741.96 
19.90 419.74 
19,877.46 4.58 
387.24 999.10 
91.85 23.46 
901.09 98 
5. $46.89 6. $48.34 
VAY 875.39 
3,538.39 82.76 
468.438 9.87 
56.19 538.49 
3:37 835.47 
786.49 3,457.96 
5,898.39 85,473.89 
65.40 4,938.78 
808.34 453.48 


—— 


- 


234 


ARITHMETIC. 


540. Supply missing numbers : 


7. 9,256,874 8. 348 
863,052 2,967 
24,635,998 36,847 
7,007,007 243,837 
? 3,096,846 
85,386,950 ee 
6,875,634 183,634 
3,987,456 986,246 
30,068 8,216 
705 586,237 
139,049,086 6,000,000 
10. 3,157,842 11. 749,809 
re 980 
1,308,215 9,876,543 
930,084 1,234,567 
17,521,938 468,208 
743,150 63,593,065 
9,807 82,389,659 
420,985 1,293,714 
73,612 460,045 
9,708 i 
65 15,813,477 
35,986,210 293,352,032 
541. Multiply: 
Lore. 20 (2xK15,014. 19. 
14, 35,482 x 7982 20. 
15. 5,290 x 6,075 21 
16. 9,204 678% 22 
LS 1 O,Uiax< 9 (395 23. 
18. 68,431 x 9242 24 


95 
185 


9. 7,293 
82,538 
786,324 

? 
94,649 
1,009,765 
256,834 
3,983,387 
54,619 
760,888 


10,685,391 


12. 8,852,465 
37,947 
40,897,654 
390,784 

? 
1,246,937 
4,373,539 
301,236 
9,764,318 
665,524 
74,638 


100,001,010 


x 95x 95 
SLU Pers 


874 x 23x 386 


. 706 


x 804 x 509 


482x 82x 74 


. 0388 X 247 x 125 


REVIEW. 250 


542. Divide: 

25. oec00= 1 810 $1. 68,703,705 + 12,345 
26. 2,823,150~— 1,298 32. 861,420,135 + 56,789 
27. Goo dae 11624 33. 70,870,088 + 25,986 
28. 21,345,738 + 72,100 34. 4,510,940+ 4,900 
29. 1,861,704= 3,510 35. 34,689,215 + 39,783 
30. 20,857,884-+ 38,004 36. 12,845,678 + 57,095 


543. Slate Problems. 


1. Thesum of three numbers is 150. Two of the numbers are 
8 and 48. What is the third? 


68 + 43+ ?=150 

2. The divisor is 24; the dividend is 264. Find the quotient. 

38. The product is 228; the multiplicand is 19. What is the 
multiplier ? 19 «? = 228 

4. The minuend is 175; the subtrahend is 87. What is the 
remainder ? 

5. The remainder is 92; the subtrahend is 89. Find the 
minuend. MD RG I2E G0 

6. The minuend is 176, and the remainder is 99. What is 
the subtrahend ? 


7. The multiplier is 15; the multiplicand is 46. What is 
the product? 


8. The multipler is 16; the product is 272. What is the 
anultiplicand ? 


9. The dividend is 800; the divisoris 17. Find the remainder. 
10. The quotient is 15; the remainder is 3; the divisor is 8. 
What is the dividend ? 
8) 2? 


153 


236 ARITHMETIC. 


11. The dividend is 273; the quotient is 21. What is the 
divisor ? 
DAE 
21 


12. The dividend is 267; the quotient is 138; the remainder 
is 7. What is the divisor? 
?)267 


132 


544. Oral Problems. 


1. What will be the cost of 8 pounds of meat at 15 cents per 
pound? 


2. Paid $12.90 for 3 pieces of lace. How much did each 
cost ? 


3. Gave $1 in payment for a 25-cent ball, and 4 ten-cent 
bats. How much change did I receive? 


4. If 3 straw hats cost 63 cents, what will be the cost of 5? 


5. At the rate of 3 oranges for 5 cents, what will be the cost 
of a dozen oranges? 


6. A gross is 12 dozen. How many pens in } gross? 
7. How many inches in 4 yards? 
8. How many ounces in 62 pounds? 


9. At 5 cents per pint, how much would be paid for a bushel 
of chestnuts ? 


10. A person used 2 gal. and 3 qt. of milk in one week, and 
8 gal. and 1 qt. the next week. How many gallons are used in 
the two weeks ? 


11. I sold 33 yards of silk and 22 yards of velvet. How 
many yards in all did I sell? 


12. A man had $64 dollars, and he spent $34. How much 
money had he left? 


REVIEW. 237 


545. Slate Problems. 


1. A man works 9 months, 26 days per month, and receives 
$702. What are his daily wages? 


2. A merchant buys 136 vases for $272. If 86 are broken, 
how much must he charge apiece for the others to gain $28 on all? 


3. On Monday, the receipts of a store are $180; on Tuesday, 
$30 less; on Wednesday, $110 less than the total of Monday and 
Tuesday. What are the receipts for the three days? 


4. The yearly rent of a house is $480. What is the rent for 
2 years 4 months? 


5. A mechanic works 300 days per year, at $4 per day. It 
his daily expenses for 365 days average $3, how much money 
does he save each year? 


6. A woman pays $5.20 for 3 lb. of tea and 56 lb. of sugar. 
What is the price per lb. of the sugar, if the tea costs 80 ¥ per lb.? 


7. A man had $7,500. He paid 2 of it for a house, $575.60 
for repairs, and $387.75 for furniture. How much money had 
he left? 


8. How much hay will be required to feed 18 horses a year 
of 366 days, if each horse receives 15 pounds a day? 


9. A person pays a debt of $576, giving 40 ten-dollar bills, 
30 two-dollar bills, 6 one-dollar bills, and the remainder in five- 
dollar bills. How many of the last did he give? 


10. A drover buys 64 sheep for $400. He sells +of them at $7 
each, and the remainder at $8 each. What is his profit? 

11. A merchant sells 56 yards of cloth for $84, gaining $14. 
What did it cost him per yard? 

12. A package of coffee, costing 60 cents, was sold for 75 cents, 
the profit on each pound being 5 cents. What was the selling 
price per pound? 


13. How many yards of cloth, at $1.75 per yard, can be 
bought for $105? 


238 ARITHMETIC. 


14. A tailor buys a piece of cloth for $50. From it he makes 
4 pairs of trousers, which he sells at $7 per pair, and 4 coats, for 
each of which he receives $15. Thread, buttons, lining, etc., 
cost him $16. How much does he get for his labor ? 


15. A man sold a certain number of papers for 50 cents. If 
he had sold 9 more, he would have received 95 cents. How 
many papers did he sell? 


BILLS. 
546. Brooxuyrn, July 31, 1894. 


Mrs. H. J. SHoRtT 
Bought of ABRAHAM & STRAUS. 


1} yd. Plaid $ 1.00 

16 yd. Cambric .05 

12 pr. Socks .20 
1 Wrapper 1 | 98 
4 yd. Silk .65 
1 pr. Gloves 2 | 25 
2 spools Silk .08 


1. Copy the above, filling in the cost of each item and the 
total. ; 


2. Robert J. Wildes buys of Caswell & Donaldson 64 |b. of 
sugar (@ 41¢; 28 lb. of lard @ 71; 24 lb. of coffee @ 25¢; 
1 bbl. flour @ $5.75; and 12 gal. of molasses@ 251”. Make 
out the bill. 


3. Make out a bill for 10 pairs of men’s shoes at $4.75; 4 
pairs of boys’ shoes at $1.474; 6 pairs of slippers at $.871; 9 pairs 
of girls’ shoes at $2.43; 8 pairs of women’s shoes at $3.374. 

4. Make outa bill for 84 lb. of ham at 14 per lb.; 34 Ib. 


of beefsteak at 24%; 9 lb. of corned beef at 12%; 101 Ib. of 
chicken at 30%; 12 lb. of roast beef at 18¥. 


DECIMALS. 239 


5. Make out a bill for 14 doz. collars, at $1.50 per dozen; 
6 doz. pairs of cutis, at $2.75 per dozen pairs; 4 doz. shirts, 
at $9.00 per dozen; 3 dozen ties, at $2.25 per dozen; 17 doz. 
pairs of socks, at $2.50 per dozen pairs. 


DECIMALS. 
547. Oral Exercises. 


In the number 25, what does the 2 stand for? 

In the number 467, what does the 4 represent? The 6? The 7? 

In the number 383,333, give the value of the first-3 (commencing 
at the left). Of the second. Of the third. Of the fourth. Of 
the fifth. 

The last 8 is what part of the number represented by the 
fourth 3? The third 3 is what part of the second? Hach 3 is 
what part of the 3 to its left ? 

The value of each 3 in this number depends upon what? 

In the number XX XIII, what is the value of the first X? Of 
the second? Of the third? 


548. When we write $784.56, the 7 stands for seven times 
how many dollars? The eight for 8 times how many dollars? 
The 4 for four times how many dollars? The 5 stands for five 
times what part of adollar? The 6 stands for six times what 
part of a dollar? 


Hundreds. Tens, Units, Decimal Point, Tenths. Hundredths. 


7 8 4 5 


This is read 784 and 56 hundredths. 

37.5 1s read 37 and 5 tenths. 

6.492 is read 6 and 492 thousandths. 

.0005 is read 5 ten-thousandths. 

01234 is read 1,234 hundred-thousandths. © 
56.000246 is read 56 and 246 millionths. 

$ 497.625 is read 497 dollars and 62 cents 5 mills. 


Norre.— Cent means hundredth. Mill means thousandth. 


240 


549. Nors. 


ARITHMETIC, 


NOTATION AND NUMERATION. 


In reading a number containing an integer and a decimal, 


the word and may be placed between the two, as is shown above. To avoid 
mistakes, the word units should be used after the integer in reading such 


numbers as 200.005, 3000.0075, etc. 


Say: Two hundred units and five 


thousandths, three thousand units and seventy-five ten-thousandths., 


550. Read the following: 


LPC § Sone 
2. 84.9 Oren a Oe 
Si) oO 10. OD 
4 eo 11. 100.025 
5. woreto 12} 125 
6. 9.624 13. .99 
Teo 14; .08 


551. Express in decimals: 


15. 005 
16. 1.3848 
17 oo.G 
18. 100.25 
19. 627.009 
20. 099 
5 ss 


887 and 72 hundredths. 


6,054 hundredths. 
6,000 and 54 hundredths. 


1. 7 tenths, 
2. 86 and 47 thousandths. 
3. One hundred twenty-five thousandths. 
4. One hundred units and twenty-five thousandths. 
5. 47 hundredths. 
6. Four hundred units and six tenths. 
7. Four hundred six thousandths. 
8. 38 and 56 hundredths. 
8. Twenty-eight. 15. 
10. 65 tenths. 16. 345 tenths. 
11. 6 and 5 tenths. shy es 
12. 465 and 8 hundredths. 18. 
13. 3875 hundredths. 19. 5 millionths. 
14. 4,000 tenths. 20. 562 millionths. 


DECIMALS. 241 


ADDITION OF DECIMALS. 


554. Add: 
cs ey 2. 3.84 3. 28.978 4. 5.6 
4.18 68.075 .28 387 
005 oO 5.375 26.93 
5.67 24.698 18.758 8.754 
10.555 97.113 


os 


© M 2 o 


10. 
7h 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 


20. 


027 + 1.89 + 48.6 + 72.978 

234.96 + .675 +50.4-+ 6.02 + 1.001 

8.047 + 54.79 + .097 + .76 + .862 

8+ 38+ .479 + 27.87 + 875 

445 + 34.75 + 306.973 + .004 + 48.56 

81+ 12.654 + 234.79 + 8.6 -+ 603 + 42.96 

45.78 + 237 + 6.987 + 18 + 372.003 + 87.5 

4.745 + 36,58 + 725.894 + 9.87 + 75.357 + 86.74 

59.8 + 83 -+9.64+4 48.565 + 6.98 + 8.795 + 963.826 
13.387 + 72.563 -+ .7-+ .603 + 7.245 + .483 + 9.25 

8.3 + 2.576 + 3.424 1.5 + 6.279 + .008 + 1.417 
24-2.35641.144 24 4.96 + 3.2724 .7 + 3.54 

4.7 4+.1.198 + .35 + 763.5 + 1,423 + 157.24 + .487-+9.5 
7.369 + 1.72 + 32.948 + .429 + .74+ 3.14 + 695+.7.005 
8.87 + 2.694 + .8 4251.47 +9.3-+ 1.916 + 41.5 + 751.006 
87 + 6.3 + .008 + 9.63 + 96 + 47.82 + 637.46 + 1.923 


242 


ARITHMETIC. 


SUBTRACTION OF DECIMALS. 


555. From 37. 182.01 
Lakeovo.t 4.624 
3434 TSG 


556. Find answers: 


21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 


1 — .057 

1 — .245 

6 — 3.324 

4 — 2.491 

3 — 1.568 

7 — 4.736 
3.587 — 1.34 


91.3852 — 72.456 
42.007 — 17.658 
68.217 — 89.4 — 


31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 


1 28.6 
009 1,008 
991 © 97.597 

9.34 — 5.672 


45.268 — 23.068 
219.843 — 187.95 
681.38 — 94.572 
1000 — 465.874 
30.053 — 18.7 
2,568.91 — 1,925 
1.234 — .825 
478.5 — 298.572 
57.088 — 44.95 


MULTIPLICATION OF A DECIMAL BY AN INTEGER. 


557. Three times 3 tenths equals how many tenths? 


Oo XO = what? Oe 4 Sie 
diy, 7.5 275 
x8 x4 Gd De 


39 
xX 20 


7.00 


1G at 
0036 
x 110 
8960 


558. The products are 13.6, 30, 3.3, 7, and .396; the ciphers to the right 
of the decimals, having no value, are omitted in giving the answers, 


359. Multiply : 


560. 


41. 
42. 
43. 
44. 
45. 
46. 
47. 


86 x 8 
ay Sai | 
6.4 x 122 
122 x 6.4 
67 x 4 
008 x 512 
512 x .008 
056 x 987 
SP aa OS) 


. 9,430 x 8 


Sight Exercises. 


Give products : 


561. 


ahs 


684 x 10 


2. 68.4 x 10 
3. 
4 
5 


Bex LU. 


» O71 X-100 
- 5.71 x-1,000 


Give quotients : 


11. 
12. 
13. 
14. 
15. 


932 + 100 
86 + 1,000 
328 + 10 
9 + 1,000 
48 + 1,000 


‘DECIMALS. © 


51. 
52. 
53. 
54, 
55. 
56. 
BT. 
58. 
59. 
60. 


© MW WD 


2715 X 1.2 
88.4 x 25 
048 x 875 
12.67 x 300 
6.57 x 9 
748 x .97 
8.76 x 43 
964 x .347 
570 x 11 
860 x .005 


.961 x 100 


57 X 1,000 » 


.09 x 1,000 
.026 x 100 
5.17 x 10 


684 = 100 
57.6 +10 
24.3 + 100 
8.75 + 10 


. 932.5+ 100 


243 


244 ARITHMETIC. 


DIVISION OF A DECIMAL BY AN INTEGER. 


562. Sight Exercises. 


1. 8.64+2 6. .846+6 
2. 48.244 7. .048+8 
3. 465-3 8. 81+9 
4. 840-5 9. 12+5 
5. 84+5 10. .24+4 


563. Where it is necessary, ciphers may be annexed to the right of the 
decimal in the dividend. 


8) .12 15) .06 1.875 
O15 .004 64) 120. 
560 
012 A418 480 
125) 1.5 21)8.673 320 
250 aT 0 
0 63 
564. Divide: 
11. 25)1.00 21.0 L= 
12. 4)21.80 22. i= 
13. 8).2 PA ye ee 
14. 13)3.913 24. tem 
15. 12)48.12 25. = 
16. 11) 70.07 26. += 
17. 24)36.6 27. 1,00 — 
18. 18).576 28. 180 — 


19. 25) 11.1 29, 5990 — 
20. 32) 62.000 $0: aks 


REVIEW. 245 


565. Slate Problems. 


31. A franc is 19.3 cents. Find the cost in United States 
money of goods amounting to 1,250 francs. 


32. A merchant bought 1,800 meters of silk. How many 
yards did he buy, a meter being 39.37 inches? 


33. A kilogram is 2.2046 pounds. What is the difference in 
weight between the English ton of 2,240 lb. and a French ton of 
1,000 kilograms ? 


34. A cubic foot of water weighs 1,000 ounces. How many 


pounds does a cubic foot of gold weigh, gold being 19.4 times as 
heavy as water ? 


35. There are 128 cubic feet in a cord. How many tons of 
2,000 lb. are there in a cord of pine wood, the latter being .66 
times as heavy as water ? 


36. A man buys three plots of ground containing 35.27, 17.8, 
and 40.375 acres, respectively. Find the total cost at $36 per 
acre. 


37. How many pints are there in 2.375 gallons? 
38. What decimal of a peck is a quart? 


39. What will be the cost of carrying 468 tons of coal at 
$0.125 per ton? 


40. A farmer sold one-eighth of his farm of 224.2 acres at 
$62.50 per acre. How much did he receive for it? 


41. How long is a post which is 5% feet above water, one- 
half of its length in the water, and one-fourth of its length in 
the mud? (Diagram.) 

42. Hight pounds of black tea costing 35% per Ib. are mixed 
with twelve pounds of green tea costing 50% per lb. What is 
the cost of a pound of the mixed tea? 

43. How many bushels, pecks, and quarts, are there in 1,449 
pounds of corn weighing 56 lb. per bushel? 


246 ARITHMETIC. 


MEASUREMENTS. 


569. Preliminary Exercises. 


Draw a square each side of which is one inch. This is called 
a square inch. 

Draw a rectangle two inches long, one inch wide. How many 
square inches will it contain ? 

Draw a rectangle three inches long, two inches wide. Divide 
it into one-inch squares. Count them. How many are there? 
How many square inches in the rectangle ? 

How many square inches in a rectangle 6 inches long, 3 inches 
wide? 

How many square inches in a rectangle 4 inches long, 4 inches 
wide ? 

About how long is your slate? About how wide? About 
how many square inches are there in its surface? 

How many square inches are there in a rectangle 12 inches 
long, 3 inches wide? Ina rectangle 1 foot long, 3 inches wide? 
In a rectangle 1 foot 1 inch long, 4 inches wide? 


570. Slate Exercises, 


How many square inches in each of the following rectangles? 
. 18in. by 14in. 5. 2lin. by 19 in. 9. 18 in. by 22 in. 
.17in.by Qin. 6. 387in. by 14in. 10. 64 in. by 29 in. 
qleun. bye. ine 9 7513 1m. Dy 4240. 9 dele Lae eames 
2010, by 15in, 8. 27 in by slin, 4 AZo itl in by rain 


fF WwW DO KF 


S71. Nore. Change each dimension to inches before multiplying. 


ASU th 1D yl an. 17a 2 it 5 Olin yt ihe a 
14. 1 ft. by 1 ft. 18. 3 ft. Tin. by 2 ft. Qin. 
15. 1 ft. 4 in. by 1 ft. 19. 4 ft. ll in. by 1 ft. 8 in. 


16. 2 ft. 6 in. by 1 ft. 20. 5ft 3 in. by 2 ft. 11 in. 


MEASUREMENTS. 247 


572. Oral Exercises. 


How many square feet in a rectangle 2 feet long, 1 foot wide ? 
6 feet long by 5 feet wide? 9 feet long by 7 feet wide? 


573. Slate Exercises. 
Find the square feet in the following: 


1. 12 ft. by 14 ft. 6. 29 ft. by 12 ft. 
2. 15 ft. by 17 ft. 7. 154 ft. by 12 ft. 
3. 19 ft. by 11 ft. 8. 15 ft. 6 in. by 12 ft. 
4. 23 ft. by 15 ft. 9, 183 ft. by 16 ft. 
5. 18 ft. by 16 ft. 10. 18 ft. 9 in. by 16 ft. 


574. Nors. Change the inches to fractions of a foot. 


11, 232 ft. by 18 ft. 16. 36 ft. by 23 ft. 5 in. 
12. 24 ft. 8 in. by 18 ft. 17. 18 ft. by 248 ft. 
13. 19 ft. 3 in. by 16 ft. 18. 13 ft. 4 in. by 24 ft. 
14, 24 ft. by 17 ft. 9 in. 19. 26 ft. 8 in. by 15 ft. 
15. 24 ft. by 16 ft. 1 in. 20. 122 ft. by 12 ft. 


575. Suggestive Examples. 


Measure the top of the desk, and calculate the surface in square 
inches. (Do not include fractions of an inch.) 

Measure the blackboard, and find how many square feet in its 
surface. (Do not include fractions of a foot.) 

Calculate the number of square inches in a pane of glass in the 
school-room window. 

Find the number of square feet in the floor of the class-room. 

Find the number of square feet in the class-room ceiling. 

Estimate the height of the class-room, and calculate the num- 
ber of square feet in the front wall. In the rear wall. In the 
right-hand wall. In the left-hand wall. 


248 ARITHMETIC. 


576. Slate Problems. 


Suacestion. When the surface is required in square inches, change each 
dimension to inches; when required in square feet, express each dimension 
in feet, or in feet and the fraction of a foot; when required in square yards, 
etc., express each dimension in yards, etc. 


1. How many square feet are there in the surface of a field 
125 feet long, 87.5 feet wide? | 


2. A rug is 2 yards long, 13 yards wide. How many square 
yards does it contain ? 


3. How many square yards are there in a strip of carpet 6 
yards long, 27 inches ($ yd.) wide? 

4. Find the number of square meters in a room 12 meters 
long, 9.75 meters wide. 


5. At 50 cents per square yard, what will be the cost of the 
oil-cloth needed to cover a floor 18 feet (6 yd.) long, 15 feet 
(5 yd.) wide? 

6. What will be the cost, at $1.50 per square yard, of car- 
peting a room 64 yards long, 15 feet wide? 


7. At 3 cents a square foot, how much must be paid for 10 
boards, each 16 feet long, 4 foot wide? 


8. A field is 80 rods long and 24 rods wide. How many 
square rods will it contain after a strip 24 rods long and 2 rods 
wide is taken from it for a road ? 


9. How many square yards of plastering will be required for 
a ceiling 18 feet long, 15 feet wide? 


10. Ifa roll of wall paper is 24 feet long and 18 inches wide, 
ow many square yards does it contain? 


CHAPTER VII. 


FRACTIONS. — DECIMALS. — BILLS. DENOMINATE NUM- 
BERS. — MEASUREMENTS. 


ADDITION OF FRACTIONS. 


577. Slate Exercises. 
1. Find the sum of 102, 6%, 57. 


Since 9 is a multiple of 3, any multiple of 9 will be a multiple of 3. 
Omitting the latter number, we find the least common multiple of 8 and 9, 
72, which is the least common denominator of the fractions. 

2. Add 24, %, 52, 44 
Omitting 4, which is a factor of 8, we find the least common multiple 


of 3, 7, 8. 
3x7xX8=168 


$. 6844478 49—7 
L. C. M.=15~x 8 Omit 5 and 3. 


4, 172-48 + 26% 


8x 9x 7=504 


epee SI -t O72 2b OF 
L. C. M. of 15 and 6 =? 


6. 29,84 45 + 16.4, 8. 114534 13,5, 4+ 82+ 198 
(Eo Sop tee tae 9. 885 -+37$ + 284 + 99% + 195 


10. iy -+ 28d, + Slat + 15,2, 
249 


250 


578. Find answers : 
11: 
1y-# 
13. 
14. 
15. 


579. 


21. 
22. 
23. 
24. 


25. 


580. Oral Exercises. 


Give two factors of : 


oO FP OW Ne 


ARITHMETIC, 


SUBTRACTION OF FRACTIONS. 


pee ee 
23,5, — 16% 
472. — 88 


102% — 848 


39,4, — 1941, 


REVIEW. 


16. 
17. 
18. 
19. 
20. 


Perform the operations indicated : 
16 x (25+ 1) 
(8h —8}) lf 
18} — (8}—-1) 


(244 — 152) x 36 


26. 
27. 
28. 
29. 


30. 


39,9, — 23.7, 

92,7, — 68,5, 
8302, — 807% 
12,345 — 6,08214 
320-9, — 120%, 


(508 +5) + 122 
100 — (4 x 188) 
(83% +172) x 24 
881 + (172 x 24) 


UX Bae Ss 
16x3 15 


FACTORS AND MULTIPLES. 


4 Ans. 2,2 6 Ans. 2,3 
Gueeo Tlie 59 16. > 57 
Tanne 12. 46 ysis. 
8. 34 13. 49 18. 62 
9. 35 it Bo tay k 19. 65 
10. 38 15. 59 20. 69 


PRIME NUMBERS. 


581. Sight Exercises. 


Give three factors of : 


Whe te: 
12 
18 
20 
27 


vee ereec mee 


oo nr & 


28 
30 
42 
44 
45 


11. 
12. 
13. 
14. 
15. 


50 
52 
63 
66 
68 


PRIME NUMBERS. 


16. 
17. 
18. 
19. 
20. 


70 
75 
18 
98 
99 


582. A number that has no factors is a prime number. 


1, 2, 3, 5, 7, etc., are prime numbers. 


583. 1. Name the prime numbers between 10 and 20. 


4. Between 50 and 70. 
5. Between 70 and 100. 


2. Between 20 and 380. 
3. Between 30 and 50. 


584. Sight Exercises. 


Give the prime factors, commencing with the smallest. 
11. 
12. 
13. 
14. 
15. 


1eLD 
2. 16 
3. 24 
4. 32 
5. 36 


6. 
7. 
8. 
9. 
10. 


40 


585. Slate Exercises. 


1. 86 
2. 87 
3.. 88 
4. 90 
6. 91 


6. 
7. 
8. 
9. 
10. 


92 
93 
94 
95 
96 


Lis 
12. 
13. 
14. 
15. 


64 
72 
74 
76 
{ire 


100 
120 
210 
240 
360 


16. 
17. 
18. 
19. 
20. 


16. 
17. 
18. 
19. 
20. 


80 
81 
82 
84 
85 


576 
840 
1,152 
1,728 
2,016 


251 


252 ARITHMETIC. 


GREATEST COMMON DIVISOR. 


586. A common faetor of two or more numbers is a number 
that will divide each of them without remainder. 

The largest number that is a factor of two or more numbers is 
called the greatest common divisor. 


587. Sight Exercises. 


Find the greatest common divisor of : 


1. 27 and 48 6. 34 and 51 
Qo Andis 7. 382 and 48 
3. 386 and 54 8. 45 and 75 
4. 26 and 39 9. 40 and 65 
5. 42 and 63 10. 54 and 69 


588. Slate Exercises. 


Reduce the following fractions to lowest terms: 


1. 14 5. 348 9. $14 
2. 278 6. 328, 10. 324 
3. 444 7. M1. $i 
4, 132 8. 24 12. 43$ 


LOWEST TERMS. 


$89. How can you tell that a number is divisible by 2? By 5? 

A number is divisible by 3 when the sum of its digits (figures) 
is divisible by 3; it is divisible by 9, when the sum of its digits 
is divisible by 9. 

A number is divisible by 4, when the number expressed by its 
last two figures is divisible by 4. 

When is a number divisible by 25? 


590. A fraction is reduced to lowest terms by dividing the 
numerator and the denominator by their greatest common divisor, 


LOWEST TERMS. 253 


591. Reduce to its lowest terms 546%. 


In this example, it is not easy to ascertain by inspection any number 
that will divide both terms. In such cases, the greatest common divisor is 
found by dividing the denominator by the numerator. The remainder is 
divided into the numerator, and each subsequent remainder is divided into 
the corresponding divisor until there is no longer a remainder. This last 
divisor is the greatest common divisor of the two numbers. 


5 
169) 10011 
156) 169° 
13) 156. 
12 


13 is the greatest common divisor. 


IGG ee La) ae 
—_—_— = — lowest terms. 
POUT eels at. 


§92. In reducing fractions to lowest terms, the above method 


of finding the greatest common divisor should not be resorted to 
if it is possible to get along without it. 


593. Reduce to lowest terms: 


123 
LL 


A look at both terms shows that 3 is a common factor. This reduces 
the fraction to 744. 41 is a prime number, and is not a factor of 100, so 
that 54), cannot be reduced to lower terms. 

2. $34 

44+3+4+2=9; 64+2+1=9 

Since the sum of the digits of each term is divisible by 9, this number is 

a common factor, and reduces the fraction to €§, etc, 


3. 44% 

5 is clearly a common factor, etc. 
4. 42 nr cam 12. 1 
5. i 9. 3 13. 119 
6. 4? 10. 5&5, 14, 115 
7. 24 11. {45 15. 248 


254 ARITHMETIC. 


LEAST COMMON MULTIPLE. 
594. Sight Exercises. 


Give the least common multiple of : 


1. 16 and 24 6 2B ORO RIO 

2. 12iandds TaD BySSG) OAD 
Bee Oa 8: 8,6, 9/1254 18 
An aso Lalo OF Dl las 
§2(2))5,450;6 10. '5,.10, 20° 26.400 


595. Slate Exercises. 

Add 141, 732, 68, %, 232, 101.8, 584, 9,4. 

Here we have to find the least common multiple of 3, 9, 7, 14, 6, 14, 2, 12. 

Rejecting 3, 6, 2 because they are factors of 12; 7, a factor of 14; and 
one 14, we have to find the least common multiple of 

9 14 12 

Divide these numbers by a prime number that is 
exactly contained in any two of them, bringing down 2). he eh 
the numbers that are not multiples of the divisor. 3} 9 7 6 

Taking 2 as a divisor, bring down 9, and write quo- 3 7 Je 
tients 7 and 6. 

3 being a factor of two of the three numbers, 9, 7, 6, is taken as the next 
divisor. 3 is written as a quotient, 7 is brought down, 2 is a quotient. 

As there is no factor common to any two of the numbers, 3, 7, 2, we 


find the least common multiple by multiplying together the two divisors 
and these three numbers. 


2%3 XK 3507-% Biss 252 LUG. Mi 
252 


Find the sum. 


FRACTIONS. 255 


596. Find the least common multiple of the denominators of 


; i Bee S28 oT 
the fractions 3, 34, 3, 44, 33, 7. 


45 
2| 20 30 45 12 Omit 4 and 6. 
= | 10 1 45 6 Strike out 15, a factor of 45. 
5 A5 3 Strike out 5 and 3, factors of 45. 


L. C. M. = 2x 2 x 45 = 180 
Add the fractions. 
597. Find the L. C. M. of: 
#40,9, 0,0, 20. Strike out 4) 3.5: 
9, 15, 15, 4, 4, 12, 25. Strike out one 15 and two 4’s. 
Poy, (, 0, 14. 10. 12° 94 
000,070,910, 1), 16,80 
20, 30, 40, 50 
2, 3, 4, 6, 8, 12, 16, 24 
24, 12, 5, 8; 10, 18 
LPB Te Fe38 
18, 5, 9, 40, 16 
LOI DAT 


CO NAA Pw DY 


rey 
= 


ADDITION AND SUBTRACTION OF FRACTIONS. 


598. In adding or subtracting fractions, they must be reduced 
to a common denominator. The least common denominator is 
the least common multiple of the denominators. 

In the following examples, determine the least common denomi- 
uator by wmspection, if possible. 


599. Add: 


IMG Veneer 6 

1. 8, 54, 34 3’ 5?) 10) 20° 25 
1 32 474 
OF 0? oF, ¢ bz. 77 


453, 201, 83, OL 
B21, 198, 61, 812 
O21 20, 88, 8, BL 
81, 458, 217, 42 


© © 2 


op wo 


1 
3 3 
4h 10. Liagoo 100) 635, 1000) 100 


>) 


“ 


56 


ARITHMETIC. 
600. Subtract: 
11. 6544-575, 16. 251.8 — 2744 
12. 1847 — 928 17. 75512 — 28325 
13. 10043 — 1513 18. 10054, — 89343, 
14. 102,8 — 2747 19. 12333 — 8018 
15. 20814 — 12838 20. 671, — 587445 
601. Perform the operations indicated : 
Dhe cabatitas 
25+5 25 
a2, 22 = 
23. (873 — 112) — (28,5 — 1934) 
24. 148 — 81 — 384 41 
25. (Bf; + 68) — Bi — 69) 
26. 48x 16 x 8% 
aT. (25 +54) + (t+ 24-484) 
28. (8}+ 41) + (2k-+14) 
29. (33; x 36) x 83 
30. 42+ 31 — 68+171 — 94 
602. Reduce to lowest terms: 
31. 5 35. 375, 39. 875 
32. x00 36. ify 40. 3125. 
33. z00 87. 825 41. ;88, 
34. slo00 38. ibty 42. 7625, 


REVIEW. 257 


603. Oral Problems. 


1. A person traveling from New York to Albany (140 miles 
apart) has gone 93 miles. How much farther has he to go? 


2. There are 196 pounds of flour in a barrel. How many 
pounds in } bbl.? | 


3. How many fourths in 24%? 

4. Reduce 4% to lowest terms. 

5. Change 19° to a mixed number. 
6. Add 4, 4, and 4. 


7. From a chest of tea containing 454 lb., 14 lb. are sold. 
How many pounds remain ? 


8. From 4 of a dollar take 102 cents. 
9. How many cents in++4- +3, of a dollar? 


10. A farmer has 602 acres of pasture and 203 acres of wood- 
land. How many acres in both? 


11. The circumference of a circle is about 3} times its diameter. 
If the diameter is 8 feet, about what is the circumference ? 


12. Mary is 12 years and 7 months old; Jane is 3 years and 
11 months older. How old is Jane? 


13. Ina year anda half William will be 7 years 2 months old. 
How old is he now ? 


14. What number multiplied by 3 equals 231? 
15. What number between 7 and 12 is a prime number? 


16. A boy received 9 marks in arithmetic, 8 in penmanship, 
and 7 in reading. What was his average mark ? 


17. 4 of a class consists of boys. How many girls in the class 
if it contains 49 pupils? 


18. If July 1 falls upon Tuesday, what will be the date of the 
third Tuesday of July ? 


258 ARITHMETIC, 


19. If July 1 falls upon Thursday, upon what day will the 
first of August fall ? 


20. A man bought 204 Ib. of sugar; he sold 10% lb. at one 
time and 64 lb. at another. How much had he left? 

21. If3 qt. 1 pt. oil cost 7 cents, what will 1 gal. 1 qt. cost? 

22. How much will have to be paid for 7 cows at $50 each, 
and 4 horses at $150 each ? 

23. %—how many hundredths? 

24. What are the two factors of 87? 

25. Find the G.C. D. of 36 and 54. 


604. Slate Problems. 

1. A merchant sold one firkin of butter at 193 per lb., a 
second at 203¢ per lb., a third at 16Z¢ per lb. What was the 
average price per lb., each firkin containing 56 lb. ? 

2. If eggs are sold at the rate of 21 for 25 cents, how much 
will be paid for 54 dozen ? 

3. Ifa train travels 45 miles per hour, how far will it go from 
half-past 9 in the morning to a quarter of 7 in the evening? 

4. To the sum of 6% and 192 add their difference, and find 
how often the greater number is contained in this total. 

5. A mechanic has deposited in the savings bank $35 per 
month for 11 months, and $20 the twelfth month. His expenses 
have averaged $3 each day of the year. What are his daily 
wages for the 300 days he has worked ? 

6. A merchant sold 4 pieces of cloth containing 274 yd., 
262 yd., 29% yd., and 281 yd., respectively. How much did he 
receive for the cloth at 96 cents per yard? 

7. Reduce 18% to lowest terms. 


8. A man has 8,5, bu. of peanuts. He puts them into bags 
holding @; bu. How many bags does he fill? 


REVIEW. 959 


9. A 160-acre farm consists of 5 fields. The first contains 17.38 
acres, the second 29.4 acres, the third 35.078 acres, the fourth 
25.875 acres. How many acres are there in the fifth field? 


10. How many seconds in 7 hours 15 minutes? 


11. Find the total cost of 2 doz. rockets at $7.50 per gross, 
3 dozen Roman candles at $9.60 per gross, and 24 doz. pin wheels 
at $1.35 per gross. (1 gross = 12 doz.) 


12. From a piece of silk that contained 28 yd., there were sold 
21 yd., 64 yd., and 133 yd. Find the value of the remainder at 
$1.20 per yd. : 


13. Three pieces of cloth bought at $2 per yard cost $150. 
The first piece measures 234 yd., the second measures 302 yd. 
How many yards in the Hind piece? 


14. Three lots of tea were sold for $330. The second con- 
tained twice as much as the first, and the third three times as 
much as the first. The third lot contained 330 pounds. Find 
the selling price of the tea per pound. 


15. What part of a person’s income remains after he spends 
4, =, and + of it? 

16. A boy loses 4 of his marbles, and he gives away + of them. 
If he has 17 marbles left, how many had he at first ? 


17. A cask of molasses contained 80 gallons. One-fourth of it 
leaked out. If the molasses cost 60 cents per gallon, what price 
must be charged for the remainder so that there will be no loss. 


18. A dealer sells 13 gross, 3% gross, 441 gross, 974; gross, and 
82 gross of lead anhiel at 36 cents per ose (she much does 
he receive for all? 


19. There are four towns, A, B, 0, and D, on a certain rail- 
road running east and west. A is 414 miles west of C; D is 394 
miles east of B; B is 221 miles west of C. How many miles from 
A to D? Make a Maree 


20. If 121 dozen rockets cost $5.75, het will 15 dozen cost? 


260 


SPECIAL DRILLS. 


605. Give sums: 


59+75 22+34+18 
67+83 19+47+430 
94438  25+435+4 26 
66+56 17+18+19 
606. Give remainders : 
134 — 75 750 — 290 
150 — 83 510 — 220 
132 — 94 630 —- 880 
122 — 56 820 — 560 


607. Give products: 


49x 2 
47x 3 
48 x 4 
43 x 5 


123 x 3 
431 x 2 
122 x 4 
don X38 


608. Give quotients : 


141+3 
192 +4 
215 +5 
276 +6 


111 + 37 
192+ 48 
215 + 48 
276 + 46 


609. Give answers: 


213 
+4 
t+4 
a+! 


wmloo  celbo 


Fat pant 


| 


= 
oes ero osibo bole 
| 
celho bl 


a 


ARITHMETIC. 


560 + 390 
270 + 280 
640 + 260 
430 + 480 


131 — 65 
123 — 84 
156 — 78 
164 — 97 


46 x 6 
34 xX 6 
38 X 7 
49x 7 


925 + 25 
875 + 25 
150 + 25 
625 + 25 


66 x 12 
84 x 13 


100 x 14 


126 x 12 


225 + 154 
315+ 421 
437 + 260 
540 + 355 


279 — 154 
086 — 263 
457 — 237 
668 — 325 


AT x 25 
25 x 86 
33 X 25 
25 X 27 


266 + 7 
296-8 
414+9 
360-8 


| 
alto ci imico cxleo 


celbo leo celbo bhme 


— 
oe) 
ex|bo 
Le) 
es|bo 


Ne) 
= 
SP oloo 


— 
s) 

olen 
| 

ae “aro 


fomcl 
=) 

bo 

orb 


REVIEW. 261 


610. Oral Problems. 


1. Find the cost of 1 lb. of tea at 75 cents, and a piece of 
ham at 56 cents. 


2. A farmer sold 58 sheep from his flock of 121 sheep. How 


many remained ? 
3. What will be paid for 8 lb. of coffee at 35¢ per lb.? 


4. A laborer received $4.88 for four days’ work. How much 
did he earn per day? 


5. At $40 each, how many cows can be purchased for 
$ 2,000? 


6. Bought 20 lb. of sugar at 5¢ per Ib., and 22 lb. of butter 
at 80%. What was the amount of my bill ? 


7. A piece of cloth measuring 311 yards was divided into 2 
equal parts. Find the length of each. 


8. How many weeks in a year of 366 days? 


9. If I pay 25 cents for 3 pounds of cherries, how many 
pounds can I buy for $1.25? 


10. Find the cost of a bushel and a peck of peanuts at the rate 
of 5 cents per quart. 


11. A farmer had 164 acres of land. How much had he left 
after selling 87 acres? 


12. Find the total number of pounds in 3 tubs of butter 
weighing respectively 25 pounds, 34 pounds, and 36 pounds. 


13. At 60¥ per lb., how much tea can be bought for $5.85? 


14. A drover paid $219 for oxen, at an average price of $73. 
How many did he buy ? 


15. What will be the cost of 20 bu. of wheat at $1.044 per bu.? 


16. At 24¢ per lb., how many ounces of butter can be bought 
for 18¢? 


262 ARITHMETIC. 


17. A woman pays $540 per year for a house. What is the 
rent per month ? ; 


18. How many weeks in 294 days? 


19. At 72 per yard, what will be the cost of 2 ft. 11 in. 
of lace? 


20. How much does a grocer receive for a barrel of flour, 196 
lb., retailed at 3 cents per lb.? 


21. If 47 men can do a piece of work in 4 days, how long will 
it take 1 man to do the same work ? 


22. Find the cost of 36 acres of land at $25 per acre. 


23. If it takes 34 yards of cloth to make a coat, how many 
coats can be made from 244 yards? 


24. How much will be paid for 84 yards of silk at $13 per 
yard? 

25. If a certain quantity of provisions will last one man 215 
days, how long will it last 43 men? 


26. How many square yards are there in a rectangular field 
36 yards long and 25 yards wide? 


613. Sight Exercises, 


Add: 
Lethe 4. 11344} 7 88464 
2. 41488 5. 78+ 941, 8. 153+ 81 
3. 92478 6. 5+ 22 9. 98452 


614. Subtract 84 from 10. 
(104 — 82) is how much greater than (10 — 84)? 


To subtract 84 from 10} mentally, we find the difference between 83 ana 
10, which is 14, and to this add }. The answer is 13. 


10. 1183—62=41438=? 
11, 142-91 =4142=% 


CANCELLATION. 263 


615. Subtract at sight: 


12. 
13. 
14. 
15. 
16. 
17. 


232 — 1954 18. 141 —83 
168 — 9% 19. 27, — 72 
184) 38 2ONMaB Ee BE 
Ph 8s OI Lae OF 
yh Laarpes 22. 43.8, — 84 
107;— 54 23. 502 — 48 


616. Slate Exercises. 
Divide without writing products (Art. 385): 


4,320 + 32 
5387 +51 


. 10,246 + 84 
. 21,3821 + 97 


1 
2 
3. 8,795 75 
4 
5 


A514 

36) 17837 
343 
197 


iy: 


6. 42,387+ 123 
7. 73,690+ 345 
8. 105,261+ 624 
9. 423,958 + 1,008 
10. 867,293 + 2,534 


CANCELLATION. 


617. Slate Problems. 
1. If 17 horses cost $4,387, what will I pay for 51 at the 


same rate ? 


618. Problems consisting of multiplication and division can be some- 
times shortened by indicating the operations, and then canceling. To solve 


the above, we indicate the cost of one horse, 


17 


3 
BBR ie 1287 BL 


et and of 51 horses, 


Since 17 is contained in 51 three times, we 
cancel both numbers by drawing a line 
through them, and we write a 3 above the 
51, The answer is $4,387 x 3 = $13,161. 


264 ARITHMETIC. 


2. If 15 eggs cost 25 cents, what will 10 dozen cost? 


25 x 10 x 12 
15 
We indicate price of one egg by dividing by 15. Multiplying this by 10 
times 12, we get the price of 10 dozen. 
In this case, 15 is not contained in any number ’ 


above the line. We divide 15 and 10 by 5, canceling OB y i %. 17 


them and writing quotients 3 and 2 alongside. 3 is con- 
tained in 12 4 times; so we cancel 3 and 12. Our Ap 
answer now is 20 cents x 4 X 2 = 200 cents, or $2. p 


3. Eighteen men can do a piece of work in 26 days. How 
long will it take 13 men to do the same work ? 
One man will take 26 days x 18. 

4. Seventeen barrels of flour, 196 lb. each, were put into 


bags holding 49 pounds each. How many bags of flour were 
put up? 


5. At the rate of 23 cents for 7 pounds, how much would 
be paid for 91 pounds of flour? 


6. A bank pays $4 interest a year on every $100. How 
much interest will be paid for 3 years on $650? 


7. At $45 per thousand for bricks, what must I pay for 
63,200 bricks ? 


8. If flour is $6 per barrel (196 lb.), what must be paid for 
a 49-pound bag? 


9. A grocer buys 84 dozen eggs, and sells them at the rate 
of 21 eggs for 25 cents. What does he receive for them? 


10. A miller buys 9,840 pounds of wheat at 90 cents per bushel 
of 60 pounds. How much does he pay for it? 


11. What will be the cost of 64 sheep, if 18 cost $198? 


12. If 18 men can do a piece of work in 42 days, how long 
will it take 21 men to do the same work? 


FRACTIONS. 265 


13. What will be the cost of 66 doz. pens at 42 cents per 
gross of 12 doz. ? 


14. A certain quantity of hay feeds 15 horses 56 days. How 
long will it feed 14 horses ? 


15. A merchant bought 9 pieces of cloth, each containing 24 
yards, for $189. What was the price per yard? 


MULTIPLICATION OF FRACTIONS. 


619. Oral Exercises. 


What is 4 of 2 fifths? Of 4 sevenths? Of 6 elevenths? 
What ist of+? Of? Of4? Of? Show by a diagram. 
What is4 of 3? Of 8? Of? Of2? 

What is$ of 2? 2of4? of}? 

What ist of 8? 2o0f3? 3 of 2? 

What isd of 4? tof? BZofhZ? F of 3? 

What is the half of 14? Of 24? Of 3}? Of 4}? 

What is one-third of 12? 2o0f14? 4of2}3? 3 of 2}? 


620. Multiply 2 by 4. 


This means to find 2 of 4. 

Since 4 of } = 44, 4 of $= x, and 2 of $= x. 

2 4=,8; that is, the product of the numerators is divided by the 
product of the denominators. 

Norr. — Cancel when possible. 


621. Multiply 2 by 3%. 


mi ie 3 
tof =f Fo PAX To 


Show by a diagram that 2 x 3 tenths = 3 fifths. 


266 ARITHMETIC. 


622. Multiply 124 by 3,4. 


Reduce the mixed numbers to improper fractions. 


35 49 119 
o xa = 88 
3 
623. Slate Exercises. 
Multiply : 
1. 2 by 96 16. 35 x 82 
2. 128 by % 17. 3% by 124 
3. § by F 18. 2x 412 
4. ¢ by g 19. $ by 3 by 44 
5. 2 by 2 20. 37 of 28 of # 
6. 3,5, by 72 21. 14x$xe& 
7. 242 by 18 22. 42 of 14 of 24 
8. 698 by 32 23. 1 of 65% 
9. 1114, by 28 24. % of 558 
10. 67 by 16 25. 61x 73 
11. 23 by 32 26. 41x dt 
12. 9 x 22 27. % of 41 x 82° 
13. 174 by 62 28. 3 of 34 x 4,1 
14. 6LXF 29. 15 x 23 X 384 
15. 41 by 84 30. 21x 21 x 21 
624. Find answers: 
31, Sx +64 - 6% 36. (82 x 21) — us 
32. (8h—21) x 8 37. 54+ 62+ 73 
33. 1 of (54 — 32) 38. 188 —32— 74 
34. (244+ 161)+8 39. 2 of $ of (844 13) 


35. (Bk +21) x (8k—2h) 40. (184 — 68) +11 


FRACTIONS. 267 


DIVISION OF FRACTIONS. 


625. Oral Exercises. 


3 fourths is contained in 4 fourths how many times? 
4 fourths + 3 fourths =? 


1 -- = how many thirds? 


626. Give answers in improper fractions. 


1S 


coho 


a: Cae 1+3=? 


Can you show by a diagram that 1+2=11=3? 


627. Divide # by 3. 
We can divide by reducing both fractions to a common denominator: 
$+ $= 29+ 21 = 20+ 21 = 29 
ad ae 27 


The following is another method: 
2 is contained in 1,3 times. In # of 1, it is contained + of 2 times, 


ris 


aT: 
628. To divide by 3, we multiply by what fraction? 


629. Divide ? by 5%. 


ys is contained in 1, 16 times. In 3, it is contained $ of 1¢ times. 
4 

Canceling, e we have 4=11. Ans. 
3 


630. To divide by 3%, we multiply by what? 
631. Can you give a rule for dividing by a fraction? 


632. N. B. Change mixed numbers to improper fractions, 


268 ARITHMETIC. 
633. Slate Exercises. 
Divide : 
1. 5 83-4 16.45 Sar ae 
2. 2+4 17. o3 + 3 
8. 10+? 18. ++ 33 
4. e+ 25 19. +6 
5. = 8 + 10 20) Oe ore 
6. gots Ah) oh ack 
7 18+5 22. 4-3 
8. 5+1% 23. 8+4 
9840-91 DAN eet ae 
10. 4,5-+17 25. 85+ 34 
11. 241 20 26. 9% + 34 
12. 68+9 27. 181-112 
13. 4-74 28. 154. + 183 
14. 11+ 4 29. 164-1381 
nes aye 30. 231+ 6§ 
634, Perform operations indicated : 


. (82 x 44) — 104 

| (183-78) x3 

, (20xH+4 

- (20+4)xs 
20+ ¥) 

| (2044) +3 

- (143 x 7) —(9 x 10f) 


42. dt X 13 X 3d X 64 


22 X 44 X 31 


REVIEW. 269 


635. Oral Problems. 
Give analysis of each : 


1. If base-balls are worth # of a dollar each, what will be the 
cost of 16 base-balls ? 


2. Paid $12 for base-balls, at 3 of a dollar each. How many 
were bought? 


3. What is the cost of 2 feet of ribbon at 30 cents per yard? 


4. Find how much a yard of ribbon is worth, if 2 yard costs 
20 cents. ' 


5. If it takes $ yard of material to make one waist, how 
many can be made from a piece containing 24 yd.? 


6. If 18 jackets require 24 yards of cloth, how much is 
needed for 1 jacket? 


7. A man had 60 acres of land. How many acres had he 
left after selling ? of his land? 


8. After spending # of his money, a person had $36 remain- 
ing. How much money had he at first ? 


9. If tea is worth $ of a dollar per pound, how much can be 
bought for 4 of a dollar ? 


10. When tea is $.50 per pound, how much can be bought 
for $.75? 


11. When silk is selling at $.75 per yard, how much can be 
bought for one-fourth of a dollar? 


12. Find the cost of a gallon of milk at the rate of 9 cents for 
3 pints. 


13. 3 of a gal. of milk costs 9%. What is the price per gal.? 
14. 2 of what number is 12? 


15. lyard and 1 foot of wire cost 16 cents. How much must 
be paid for a yard? 


270 ARITHMETIC, 


16. A man bought some cows at $35 each, and the same num- 
ber at $45 each. What was the average price ? 

17. A girl received 100 per cent in three studies, and 80 per 
cent in the fourth. What was her average ? 

18. A square floor contains 144 square feet. How many feet 
long and wide is it? 


19. Mr. Brown mixed 3 pounds of black tea worth 40 cents a 
pound with 1 pound of 60-cent green tea. What is the mixed 
tea worth a pound? 


640. Slate Problems. 


1. A milliner sells 3 pieces of ribbon at 18 cents per yard. 
They measure 42 yd., 34 yd., and 535, yd., respectively. What 
is the amount of her bill? 

2. How many feet and inches in 55 of a yd.? 

3. To make powder, a man mixes 7+ lb. of saltpetre, 14% lb. of 
sulphur, and as much charcoal as sulphur. How many pounds 
of powder will there be? 

4. Four men form a partnership; the first furnishes 4 of the 
capital, the second 8, and the third 53. What fraction of the 
capital is furnished by the fourth ? 

5. I pay 15 cents more for a half pound of tea than I pay for 
a quarter pound of the same tea. What is its price per pound? 

6. After doing # of a piece of work, a man requires 3 days 
more to finish it. How many hours does he take to do the 
whole work if he works 8 hours per day ? 


7. If 1 lb. 7 oz. coffee cost 46 cents, what will 3 lb. 9 oz. 
cost ? j 


8. Add 14 days 6 hours 50 minutes and 15 days 17 hours 
10 minutes. 


9. If a dozen pairs of gloves cost $15.25, what will be the 
cost of 60 pairs? Cancel. 


REVIEW. ee 


10. 15 men doa piece of work in 102 days. How long would 
it take 5 men to do the same work? 


11. To make a cloak 3 yd. of cloth 11 yards wide are re- 
quired. How much cloth # yd. wide would be required? 


12. In3 years 4 months a gas company manufactures 4,200,000 
cubic feet of gas. How many cubic feet are manfactured per 
year ? | 

13. If 22 dozen hats cost $80, what will be the cost of 3 hats? 

14. A boy hires a boat at 20 cents per hour. How much has 


he to pay if he uses it from 20 minutes before 9 a.m. until 10 
minutes past 1 P.m.? 


15. A and B kill an ox. A takes $8 and B the remainder. 
If B’s share weighs 3614 lb., what is the weight of the ox? 

16. A grocer buys 30 dozen eggs at 18 cents per dozen. He 
sells them at the rate of 15 eggs for 25 cents. What is his profit? 


17. How many cents in 55 of a dollar? 
2 
18. What fraction of 18% 1s 62? ~% =f 


19. A farmer buys a horse for $140, and sells it at an advance 
of 8, of the cost. What is the selling price? 


20. In 1893, A was 36 years old and B was 1$ times as old. 
In 1884, B was how many times as old as A? 


21. From the sum of 18,4, and 252 take their difference. 


22. If 23 acres of land cost $220, what will be the cost of 174 
acres? Indicate the work, and cancel. 


23. A can do a piece of work in 6 days, B can do it in 6 days, 
C can do it in 6 days. How long will it take all three working 
together ? 

24. Find the value of ares 

25. A man sold a horse for % of its cost, losing $40. What 
did the horse cost him ? 


OH ie ARITHMETIC. 


FRACTIONAL PARTS OF A DOLLAR. 


641. Oral Problems. 
1. How many 50-cent base-balls can be bought for $15? 


(15 +4) 
2. How many 75-cent base-balls can be bought for $15? 
(15+ 4) 


3. At 75¥ per lb., how much tea can be bought for $1? 
4. How many hats at $1.25 each can be bought for $15? 
(15 + 14) 


5. Paid $16 for coffee at 25% per lb. How many pounds 
were purchased ? 


6. At 331% per lb., how many pounds of butter can be 
bought for $32? 


7. Find the number of yards of ribbon, at 121 ¢ per yd., that 
will cost $45. 


8. At 619 per bar, how many bars of soap will cost $11? 


9. If 4 pieces of violet soap are svld for 25f, how many can 
be bought for $9? 


10. $24 is paid for corn at 75% per bu. How many bushels? 


11. I spent $30 for lace at 662 ¢ per yard. How many yards 
did I buy? 


12. For $36 how many pairs of rubber shoes can be bought 
at 871 per pair ? 


13. Oats are 621% per bu. How many bushels will $40 buy? 


14. A farmer pays 873% per bu. for seed rye. If his bill 
amounted to $21, how many bushels did he buy? 


FEDERAL MONEY. pags: 


15. A store-keeper sold $33 worth of collars, at 162% each. 
_ How many did he sell? 


16. At the rate of 3 for 50%, how many collars can be bought 
for $25? 

17. Corn is worth 20¢ per can. How many cans will cost $32? 

18. Find the cost of 35 yards of cloth, at $1.25 per yard. 


19. At $1.25 per yard, how many yards of cloth can be 
bought for $35? 


20. How many pairs of gloves, at $1.75 per pair, will cost $28? 

21. When coal is $5.25 per ton, how many tons can be bought 
for $42? 

22. Cost of 16 pairs of shoes at $2.75? 

23. 33 jackets at $3.334? 

24. 18 yd. cloth at $2.162? 


25. Paid $26 for cloth at $2.162 per yard. How many 
yards did I buy? 


26. Find the cost of 16 pairs of skates at $1.874 per pair. 
27. Ifsheep cost $3.124 each, how many can I get for $75? 
28. How many 25-cent balls can be bought for $8.75? 

29. Divide 775 by 25. 

30. Divide $8.25 by 75 ¥. 


31. How many square feet are there in a lot 96 ft. long 25 ft. 
wide? 


32. Find the total cost of 82 head of cattle at $75 per head. 

33. How much must be paid for 32 cows at $37.50 each? 

34. If sheep are worth $3.75 each, how much will a farmer 
receive for 32 sheep? 

35. Ifa train goes at the rate of 25 miles per hour, how many 
hours will it take to go 675 miles? 


274 ARITHMETIC. 


BILLS. 


642. New York, Oct. 1, 1894. 
Mrs. WILLIAM MARTIN, 


Bought of GRAY AND WINTER. 


1894 
Aug. | 13 | 44 yd. Carpet $ .90 
15 | 3 Oak Chairs 1.75 
1 Rocker 12 | — 
19 | 18 yd. Oil Cloth .50 
27 | 1 Parlor Suit 75 | — 
Sept. | 19 | 6 Kitchen Chairs 75 
1 Table 4 | 50 
26 | 36 yd. Matting 331 


1. Copy the above. Supply the missing amounts. 


2. John R. Schultz has bought the following goods of Arthur 
B. Rowe & Co. : 

Jan. 8, 1894, 50 lb. of sugar, at 54%; 4 Ib. of tea, at 622%. 
Jan. 4, 10 lb. of coffee, at 824%; 2 bbl. of flour, at $5.75. Jan. 9, 
24 bars of soap, at 162 ¢. 42 |b. of starch, at 8¥. 

Make out a bill dated Feb. 1, 1894. 


3. Make out a bill for the following articles bought during 
March and April. Supply the names of buyer and seller, also 
the dates. 


231 yd. of silk, at 80%; 18 yd. of lace, at $2.40; 64 yd. of — 


muslin, at 62%; 8 spools of sewing silk, at 7%; 4 pr. of stockings, 
at 65%; 6 yd. of linen, at 871¢; 4 doz. collars, at $2.10. 


4. Make out a bill for the following goods bought June 15. 

3 cases of torpedoes, at $2.20; 12 boxes of fire-crackers, at 
$1.621; 3 gross pin wheels, at $1.35; 5 gross sky-rockets, at 
$3.25; 2 doz. balloons, at $2.25; 45 lanterns, at 9. 


ee stilted 


647. Sight Exercises, 


1 ie 


2, 31 


3. 12x — 


4. 


36 x 14 
9 


x 16 
8 


29 X 18 
36 


5. 


8. 


REVIEW. 
42x 28 
——— 9. 
pal 
4. 
a X46 10. 
23 
67 
Bo 11. 
96 
Bo Le 
ede ee 12, 
99 


648. Visitors to Prospect Park. 


Month. 


January . 
February 
March 
April . 
May 

June . 
July 
August 
September 
October . 
November 
December 


Carriages, 


43,398 
112,140 
128,520 
120,240 
359,621 
208,096 
220,860 
260,516 
333,639 
421,220 
316,020 
174,256 


Equestrians. 


1,953 
8,032 
12,027 
8,827 
15,805 
14,687 
5,575 
9,578 
11,926 
16,246 
15,324 
12,157 


Pedestrians. 


408,230 
485,990 
526,270 
655,925 

1,944,353 

1,233,873 

1,443,173 

1,640,651 

1,704,611 

1,699,851 

1,268,101 
719,569 


13. 


14. 


15. 


16. 


Sleighs, 


Totals. 


6,680 


16 


ee, EEE ed en, 


Monthly average. . 


Daily average . 


ee Sf | 


| | [| 


In the foregoing table find the total number of visitors for each 
month, the number of visitors by carriage for the year, the 
number of equestrians and pedestrians, the number by sleighs, 


together with the grand total for the year. 


and the monthly average. 


Find also the daily 


276 ~ ARITHMETIC. 


SHORT METHODS. 


649. Oral Problems, 
1. Multiply by 25: 
16, 19, 21, 238, 25, 29, 33, 36, 42, 48. 
2. How many square feet in a lot 84 ft. long, 25 ft. wide? 
3. What is the weight of 25 bbl. of flour, each weighing 
196 lb.? 
4. Find the cost of 25 lb. of coffee at 82¢ per lb. 
5. What will a woman have to pay for 25 yd. of silk at 
$1.60 per yd. ? 
6. A man sold 25 cows at $44 each. How much did he 
receive for them? 
7. Multiply 64 by 123. 
8. Find the cost of 124 bu. of wheat at 96% per bushel. 
9. At $12.50 per bbl., how much would I have to pay for 
56 bbl. of pork ? 
10. How many pens in 124 gross? (144 to gross.) 
11. Find the cost of 124 lb. of tea at 56¢ per lb. 


12. How many square yards in a field 96 yards long 75 yards 
wide ? 


650. Blackboard Exercises. 


Write only the answers : 


1. 887x 25 8. 25x 2,174 15. 124 x 1,084 
Ree Oo axe EAD 9. 837 x 250 16. 123 x 2,196 
3. 9384x 25 10. 763 x 250 17. 123 x 3,670 
4. 508x 25 11. 864x 124 18. 123 x 6,281 
5. 25x 686 1 A a 2 19. 864 x 125 
Cee OU 13. 236xX 122 20. 776 xX 125 
7. 25 x 1,089 14, 404 123 21. 125 x 1,020 


—s 


FRACTIONS. 


651. Add 152 and 83. 


Adding 2 and ? (or 


22. 
23. 
24. 
25. 
26. 
27. 
28. 


652. 


36. 
37. 
38. 
39. 
40. 
41. 
42. 


3414. 15,7, 
4238 + 194 
842 + 188 
404 + 162 
158 + 822 
352 + 201 
128 + 411 


10 + 
1 


29. 
30. 
31. 
32. 
33. 
34. 
35. 


Write answers: 


57} — 182 
985 — 561 
463 — 192 
675 — 8h 

745 — 401 
572 — 18} 
981 — 568 


43. 
44. 
45. 
46. 
47. 
48. 
49. 


653. Multiply 18% by 4. 


8x4=3. 4 eights are 32, and 3 are 35 (put down 5). 4 ones are 4 
and 3 are 7. 


50. 
51. 
52. 
53. 
54. 
55. 
56. 


274 x 10 
334 X 12 
168 x 8 
172 x8 
192 x 6 
153 x 3 
13% x 4 


57. 
58. 
59. 
60. 
61. 
62. 
63. 


5 *). we get ly. Write +4 and carry 1. 


O51 4+ 468 
578 +178 
29% + 844 
68% +188 
74,5, + 18% 
871 + 60$ 
138 +814 


372 — 291 
58,8, — 502 
242 — 61 
902; — 182 
6511 — 94 
872 — 292 
O41 — 6 


Ans. 75. 


204 x 11 
402 x 5 
163 x 7 
374 X 3 
458 X 5 
234 x 4 
17i x 6 


277 


278 ARITHMETIC. 


654. Do not change dividends to improper fractions. 


64. 3)453 71. 6) 25} 
65. 4)564 72. 7)10% 
66. 12)361 73. 6)752 
67. 5) 724 74. 7)97335 
68. 11)833 75. 10)874 
69. 8) 374 76. 4)662 
70. 9)481 77. 8)941 


MULTIPLICATION OF DECIMALS. 
655. Oral Exercises. 


3 and a decimal multiplied by 2 and a decimal gives about 
what product? 


44.02 x 2.05 = about what? 


656. Slate Exercises, 


Multiply : ; 
1) O26 2,0 6 9.6 x 1.125 
aoe Oi 7 34.9 x 2.34 
8. 64x45 8. 5.625 x 8.4 
4. 7.23.75 9. 1.875 x 128 
By 12.8 x 5.7 10. 42.36 x 2.95 


657. In multiplying 32 by 2.5, how many decimals are pointed 
off in the product? In multiplying 3.2 by 2.5, how many are 
pointed off? How many are pointed off in the product of 9.6 
by 1.125? 


DECIMALS. 279 


658. Can you tell the relation the number of decimal places 
in the product bears to the number in the multiplier and in the 
multiplicand ? 


qo Xqo=? ies tie chix hee abe el hiearg 


11. 1.75 x 64 16. 18.4 « 20.25 
12. 8.875 x 40 Le MEDIO A240 
18. 24.5 18.2 18. 66.6 x 3.34 
14. 96x12} 195 204 Sc TD 
15. 7.48 x 3.6 20. 400.04 x 89.25 


DIVISION OF DECIMALS. 
659. Divide 42 by 2.1. 


Changing the decimal fraction in the divisor to a common 
fraction, we have 


= 4221 — 42 10 — 420 
42 + 23, = cy MBean ROA: IG ‘ 


42 + 2.1 = 420 + 2l. 


— 


660. When we change the divisor 2.1 to 21, we have multiplied it by 
10, and the same change must be made in the dividend. 


661. In the following examples, make each divisor a whole 
number by removing the decimal point, and make a correspond- 
ing change in the dividend. 


662. Divide: 

21: 80 -- 2.5 30. 72.5 
ae 8+2.5 31. 960+ .03 
23. 840-12 32. .847+ .007 
24. 36 + 3% 33. 27 + .002 
25. 36 + .9 34. 10—18 
26. 126-638 35. 1.263 + .03 
27; 48-15 36. 196.8 + .018 
28. 18.36 + .6 37. 19.68 + .013 


29. 50 + .25 38. 1.963 + .013 


280 ARITHMETIC. 


663. Remove the decimal point in the divisor Ans. 15 100. 
three places to the right, and make a corresponding 6013.) 1966300. 
change in the dividend, adding two ciphers. ayy 

To show where the decimal point originally be- err 
longed, it may be enclosed in a small circle, instead —— 
of being erased. 00 

When the divisor is thus made a whole number, the decimal point in the 
quotient will be placed under (or over) the new decimal point in the 
dividend. 


1.736 + 16 17.36 + .16 017386 = 1.6 

1085 Ans. 108.5 Ans. .01085 Ans, 
16) 1.736 @16.) 17636. 1e6.) @0.1736 

186 186 136 

~ 80 80 80 
39. .504+ .024 47. 392 3.2 
40. 504+ .24 48. 48 + 3,009 
41. 504+24 49. 92 -- .23 
42. 504+ 24 50. .875+125 
43. 168+.7 51. 381.17+ 8.11 
44. 86 +112 52. 624 -+ 9.75 
45. .875+.25 53. 48.195 = 3.57 
46. 123.6~ .01 54. 829.31 + .019 

664. Divide 381.6 by 95.032. 
4.015 + 
950382.) 381.600. 
147200 
521680 


The sign (+) after the last figure of the quotient indicates that there is a 
remainder. 


665. Divide, carrying out the quotient to 3 places of decimals : 
5D. 31+ 13 58. 7.049 + 1.6 
56. 4.5+17 59. 81.22+ 3.275 
57. 920.07 = 46 60. 246.3 + 93.473 


Se eo ae 


13. 
14. 
15. 
16. 


666. Sight Exercises. 


Write answers at sight : 
.042 x 200 
13 x 800 
014 x 50 
8.1 x 60 


5. 


6 
7. 
8 


DECIMALS. 


40x.7 


25 X .08 


284 X .2 
.13 X 30 


667. Remember that 369 + 1,000 
= .569; that 219 100 =2.19;: and that 6 + 100 — .06. 


2,460 + 3,000 


369 + 3,000 
219 3800 
48.6+ 60 
189+ 90 


668. Slate Exercises. 


ae 
1S: 
19. 
20. 


196 + 4, 


Gin 
27.9 + 


281 


9. .121 x 4,000 
10. .061 x 500 
LE. 7 038:.2.000 
LZ Ol Byrn 100 


= 369. — 369 thousandths 


1000 


000 
500 
300 


21. 4.68-= 20 
22. 380.5+ 500 
23. 188+ 200 
24. 248+ 4,000 


Cancel the ciphers in the divisor, and remove the decimal 
point in the dividend a corresponding number of places to the 
left, prefixing ciphers if necessary. 


1. 1,728 + 1,200 
1299) 17.286 
1.44 Ans. 
4. 2.486 + 3,000 
5. 1386.5 + 1,300 
6. 848+ 80 
7. 100.1+ 700 
8. 1+ 40 
9. 22+ 50 


2. 172.8 + 1,200 


1200) 1.7268 
144 Ans. 


10. 
ul 
. 845.6 + 1,200 
. 4004+ 110 


3. 1.728 + 1,200 


1200) 016728 
.00144 Ans. 


45~ 800 
OF nat) 


5.28 eto0 


. 907.5 + 1,500 


282 


ARITHMETIC, 


SIGHT APPROXIMATIONS. 


669. Give approximate answers. Whole numbers, 


670. 


671. 


ihe 


2. 
3. 
4. 
5. 
6. 


17,8, X 388; or, about 17 Xx about 4. 


8 
9 
2075 + 24; or, about 25+ 4 nearly. 


63 X 63 7. 79934 + 9918 
3001, + 1138 8. Ta x Tp 
863 x 4 Oo Ted tte 
35% + 355 10. 64,8 


Give answers in whole numbers: 


er 


8.75 x 9.999; or, 8.75 x 10 nearly. 
24.002 + .4999; or, 24 nearly 55, or 4. 


25.125 x 11.834 Toh LO. ee 
36.845 + 6.105 8. 7.999 X:7.999 
86.4 = .983 9. 7.001 x 12.003 
82.04 x 5.001 10. 64.001 + .249 


Give the cost, approximately, of : 
1. 49 horses at $199 each. ($200 x 49.) 


_ 
= 


199 yd. 2 ft. 11 in. of cloth at $2.50 per yard. 
3 lb. 15 oz. of butter at 25 ¢ per lb. 

398 coats at $12 each. 

7 bu. 3 pk. 7 qt. potatoes at $2 per bushel. 
798 base-balls at 25, cents each. 

19 gal. 3 qt. 1 pt. alcohol at $2.49 per gallon. 
995 lb. tea at 59% cents per pound. 

7 houses at $4,995 each. 

597 pounds of hay at 99 cents per 100 pounds. 


SLATE PROBLEMS. 283 


672. Slate Problems, 
1. Find the cost of 18,756 ft. of lumber at $30 per 1,000 ft. 
2. A field is 14.25 rods long by 7.4 rods wide. What is its 


area in square rods? 
3. Arodis 16.5 feet. How many rods are there in 231 feet. ? 


4. How many marks are there in $100? (A mark is equal 
to 23.8 cents.) 


5. Add3and 4 tenths, 96 thousandths, 100 and 5 thousandths, 
27 hundredths. 


6. From 2,700 take 27 hundredths. 
7. Multiply 8 and 4 tenths by 9 and 25 hundredths. 
8. Divide 96 and 75 hundredths by 322 and 5 tenths. 


9. A load of hay, at 75 cents per 100 pounds, cost $13.98. 
What was the weight of the hay? 


10. The circumference of a circle is 3.1416 times the diameter. 
flow many inches in circumference is a circle whose diameter is 


20 inches ? 

11. Show by a diagram the number of pieces of wire ? yd. 
long that can be made from 4 yd. of wire. 

12. Show by a diagram that three-fourths of one is equal to 
one-fourth of three. 

13. If 2of a yd. of material will make an apron, how many 
half aprons can be made from a yard? Show by a diagram. 


14. A boy paid 6 cents for three-eighths of a pie. What would 
be the cost of the whole pie at the same rate? Make a drawing. 


15. Seven-eighths of an acre 
of land is sold for $140. What 
is the price of an acre? 


284 


ARITHMETIC, 


REVIEW. 


676. In comparing two fractions, reduce both to a common 
denominator. Change denominate numbers to same denominate 


unit. 


677. Oral Exercises. 


1. 


_ 
= 


11. 
12. 


Oi eet Cle Ot wy Ot 


What part of 7% 1s 33? (What part of 7 is 3?) 

4 is what part of <4? 

What part of 24 is 4? 

3 pt. is what part of a gallon? (8 pt. is what part of 8 pt. ?) 
What part of a gallon is 1 qt. 1 pt.? 

Divide 2 by 3. (Divide 10 fifteenths by 9 fifteenths.) 
Divide 3 by 2. 

How many sq. ft. in a rectangle 12 ft. long, 13 ft. wide? 
4 of a day is how many hours and minutes? 

14 ounces is what part of 2 pounds? 

$ ft. is what part of a yard? 

A strip of tape 3 yards long is cut into four equal pieces. 


How many feet and inches in each piece? 


13. At $30 per month, how much rent will I pay in 1 year, 
8 months? 

14. 24 months is what part of a year? 

15. At #2 ofa dollar per lb., how much tea can I get for $1? 

16. How many sq. yd. in a room 15 ft. long, 18 ft. wide? 

17. A lot is 25 ft. by 100 ft. How many feet of fence will 
it take to enclose it ? i 

18. 1 pk. 1 qt. is what part of a bushel? 

19. 15 is what part of 4 dozen? 

20. Reduce 2% to lowest terms. 


> —— 


LONG MEASURE. 285 


DENOMINATE NUMBERS. 


678. Slate Exercises. 
1. Add 4 days 6 hours, 9 days 11 hours, 3 days 7 hours. 
2. What part of a week is 1 day 18 hours? 


3. Ifa man receives $60 interest per year, how much will he 
receive in 3 years 74 months? 


4. Reduce 3 days 18 hours to minutes. ° 

5. How many days and hours are there in 8,100 min.? 
6. 34°) of a day is how many hours? 

7. How many hours and minutes in .4 day? 


8. A man receives $1,460 per year of 365 days. What is 
his salary per week ? 


9. Find the cost of 1 bu. 1 pk. 1 qt. of potatoes at 8 cents 

per half-peck. 

10. A piece of meat weighing 27 lb. 12 oz. is divided among 
6 persons. How many pounds and ounces does each receive? 

11. How many bu., pk., and qt., are there in 5 bags, each 
containing 1 bu. 1 pk. 1 qt.? 

12. How many gallons, quarts, and pints of ice-cream will be 
needed to give a half-pint to each one of 67 persons? 

13. Find the cost of 7 lb. 10 oz. of tea at 40 cents per lb. 


14. From a pile of 20 bu. wheat there were sold 10 bu. 3 pk. 
7 qt. How much remained? 


679. Long Measure. 


12 inches (in.) 1 foot (ft.) 

3 feet 1 yard (yd.) 

5} yards 1 rod (rd.) 
320 rods 1 mile (mi.) 


286 ARITHMETIC. 


15. How many yards in a mile? How many feet? How © 
many inches? 


16. A field is 16 rods long, 12 rods wide. How many square 
yards does it contain? How many rods of fence will be needed 
to enclose it? How many feet? 


17. How many rails each 30 feet long will be needed for a 
single track road (two tracks) 40 miles long? 


18. A boy steps $3 inches. How many steps will he take in 
going 2 miles? 


19. Dec. 20 the sun rises at Boston at 7.26 a.m. and sets at 
4.30 p.m. How long is it between sunrise and sunset? How 
much longer is the day at Charleston, 8. C., where the sun rises 
at 6.58 A.M. and sets at 4.57 p.m.? 


20. On June 21 the sun rises at Boston at 4.23 a.m. and sets 
at 7.40 p.m. On the same day it rises at Charleston at 4.53 a.m. 
and sets at 7.11 P.m. What is the length of the day at each place? 


Change: 
21. 17 lb. and 4 oz. to ounces. 
22. 84 tons and 1,560 lb. to pounds. 
23. 37 gal. and 3 qt. to quarts. 
24. 45 gal. to pints. 
25. 63 qt. and 1 pt. to pints. 
26. 27 bu. and 3 pk. to pecks. 
27. 48 pk. and 7 qt. to quarts. 
28. 84 pk. to pints. 
29. 27 mi. to yards. 
30. 16 rd. and 3 yd. to yards. 
31. 15,000 min. to days, etc. 
32. 25,124 lb. to tons, etc, 
33. 1,650 ft. to rods. 


41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 
49. 
50. 


51. 
52. 
53. 
54. 
55. 


34. 
35. 
36. 
37. 
» 38. 
39. 
40. 


DENOMINATE NUMBERS. 287 


876 pt. to gallons and quarts. 
228 in. to yards and feet. 
1,650 rods to miles and rods. 
864 hours to weeks and days. 
296 qt. to bushels and pecks. 
315 oz. to pounds and ounces. 


743 months to years and months, 


& ft. 6 in. + 9 ft. 5 in. + 12 ft. 3 in. 

380 min. 15 sec. + 80 min. 18 sec. ++ 45 min. 24 sec. 

9 yr. 35 mo. + 18 yr. 7 mo. + 22 yr. 2 mo. 

19 wk. 4 da. + 7 wk. 5 da. + 8 wk. 

9 mi. 169 rd. + 84 rd. + 3 mi. 67 rd. 

7 yd. 1 ft. + 83 yd. + 19 yd. 2 ft. 

18 gal. 1 gt. + 16 gal. 2 qt. + 15 gal. 3 qt. 

5 pk. 3 qt. +6 qt. +7 pk. 1 qt. 

24 bu. 3 pk. + 24 bu. 3 pk. + 24 bu. 3 pk. 

12 qt. 1 pt. + 12 qt. 1 pt. + 12 qt. 1 pt. + 12 qt. 1 pt. 


12 qt. 1 pt. x 4. 56., Qmt..25 rd, x. 7. 
24 bu. 3 pk. x 3. 57. 15 wk. 3 da. x 5. 
5 pk. 8 qt. x 9. 58. Tyr. 3 mo. X 10. 
18 gal. 1 qt. x 8. 59. 40 min. 35 sec. X 2, 
33 yd. 1 ft. x 6. GOs,0 ibe ine La. 

61. 25 ft. — 18 ft. 7 in. 

62. 50 min. 13 sec. — 27 min. 30 sec. 

63. 12 yr. 1 mo. — dyr. 11 mo. 

64. 50 wk. 4 da. — 18 wk. 6 da. 


65. 


15 mi. — 8 mi. 148 rd. 


288 


7a. 
72. 
73. 
74. 
75. 


ARITHMETIC. 

66. 33 yd. 1 ft. — 18 yd. 2 ft. 

67. 240 gal. 1 gt. — 94 gal. 2 qt. 

68. 83 pk. 3 qt. — 59 pk. 1 qt. 

69. 170 bu. 1 pk. — 85 bu. 2 pk. 

70. 1385 qt. 1 pt. — 67 qt. 1 pt. 
87 qt. + 2. 76. 253 yd. 1 ft. + 10. 
50 min. 85 sec. + 5. 77. 387 gal. + 6. 
156 yr. 9 mo. + 9. 78. 222 bu. 3 pk. + 9. 
73 wk. 2 da, + 3. 79. 150 qt. + 4. 
50 mi. 185 rd. + 7. 80. 75 bu. + 8. 

81. 87 qt.+ 48 qt. 1 pt. 

82. 50 min. 35 sec. + 10 min. 7 sec. 

83. 78 bu. +9 bu. 3 pk. 

84. 5 lb. 1 oz. +9 oz. 


85. 


14 ft. 2in. +1 ft. 5 in. 


MEASUREMENTS. 


680. How many square yards in a room 6 yards long, 5 yards 


wide? 


How many square yards in a room 18 feet long, 15 feet wide? 


681. Slate Exercises. 


Calculate the number of square yards in the following. First reduce each 
side to yards. 


1. 


2 
3 
4. 
5 


18 yd. by 21 yd. 
. 54 ft. by 63 ft. 

. 72 in. by 108 in. 
19 yd. by 47 yd. 
. 67 yd. by 89 yd. 10. 54 in. by 72 ft. 


38 ft. by 36 yd. 
27 ft. by 96 ft. 
54 ft. by 72 in. 
48 ft. by 45 ft. 


oO OM 2H 


MEASUREMENTS, 289 


682. First, indicate the operations; then cancel. 
11. Find the number of square yards in a room 18 ft. 4 in. 
long, 22 ft. 6 in. wide. 


18 ft. 4 in, = 182 ft. = vs yd. = - yd. 


22 ft. 6 in. = 224 ft. = =! yd, = & yd. 
5 
Area = a x ° sq. yd. Canceling, re = ge = 453 sq. yd. 
12. How many square yards in a room 13 ft. 1 in. long, 27 ft. 
wide ? 


18 fe.1in. = 157, in, = oa yd, 27 ft. =9 yd. 
157 157 x 9_ 157 
Area = ( 15” y 9) sq. ya. eae yd, 
rea C ) sq y 36 ; 394 sq. yd 
4 


13. How many square inches in 12 panes of glass, each 5 
inches long, 7 inches wide? 


14. A piece of cloth is 48 yards long, 24 inches wide. How 
many square yards does it contain? 


15. A merchant imports 8 pieces of cloth, 36 yards to the 
piece. How many square yards of cloth are there, if it is 32 
inches wide? 


16. A board fence 6 feet high surrounds a lot 25 feet front by 
100 feet deep. How many square feet of boards in the front 
fence? Inthe back fence? In each side fence? In the whole? 
(Make diagrams.) 


17. A room is 18 feet long, 15 feet wide, 12 feet high. How 
many square feet in the floor ? 


Draw a rectangle to represent the ceiling. Write the dimensions in their 
proper places, and write in the centre the number of square feet in its sur- 
face. Draw diagrams of the four walls; give dimensions and surface of each. 


290 ARITHMETIC. 


18. How many faces has a cube? If one edge of a cube meas- | 


ures 4 inches, how many square inches in the entire surface? 


Suppose you wish to make a cube out of a single piece of pasteboard. 
Make a drawing to show the shape of the piece needed, without allowing 
anything for overlapping parts. 


19. The United States government charges a duty of 4% per 


square yard on imported cotton cloth. What duty must the. 


importer pay on a piece containing 24 yards, # yd. wide? 


20. What will be the cost at $1 per square yard for flagging 
a sidewalk 12 feet wide and 30 feet long? 


CHAPTER VIII. 


DECIMALS, — BILLS. — DENOMINATE NUMBERS.— MEASURE- 
MENTS. — PERCENTAGE, — INTEREST, 


DECIMALS. 


685. Changing Oommon Fractions to Decimals. Slate Exercises. 


Reduce the following common fractions to decimals; 7.e, per- 
form the indicated division : 


1. 1+ 800 8. 3 15. 345 
2. 1+ 40 9. z2ty 16. ;is 
3. ae ip eee 17. 315 
4, 25 ieee 18. 13, 
ayo te toe tos 
6.8, 13. 23 20 ant 
7. so 14. +f 21. ine7 


686. Changing Decimals to Common Fractions. 


What is the denominator of a decimal fraction ? 

What prime numbers are contained in10? What are the only 
factors of 10? The prime factors of 100? Of 1,000? 

Can zo, be reduced to lower terms? Why? Can 78, be 
reduced to lower terms? Why? Can 7495, be reduced to 
lower terms? How can we tell by merely looking at a decimal 
whether or not it can be reduced to a common fraction of lower 
terms? 

291 


292 


ARITHMETIC. 


687. Slate Exercises. 


Reduce the following to common fractions — lowest terms. Do 
not find the greatest common divisor. 


22. 0076 BPR Uva 
rey ae 33. .027 
24. .0275 34. .00365 
25. .44 35. ..90 
26. .03125 36. .0009 
27. .486 37. .816 
28. .3750 38. .15625 
29. .37500 39. .0375 
30. .144 40. .00625 
31. .0006 41. .096 


ADDITION OF DECIMALS. 


688. Add the following, reducing the common fractions to 


decimals : 
42. 
43. 
44. 
45. 
46. 
47. 
48. 
49. 
50. 
51. 


183 + 9.084 + 25,1, + 163 + 2.09 + 86,1, + .0975 
275-95 + 58.64 + 8.6796 + 301 + 82 + 99 + 68723 
S47, + I3z hg + Syed + r3bby + 684.1 + $4174 
250 + 1875.93 + 163 + SA, + o%& + 608.94 + .0005 
8.6796 + 96.8 + 188 + 250,1, + 341, + 1876 
40,8 + 7.2832 + 86.3 i 128.46 + 23, + 41.5 + 84 
540 + 1.82 + .576 + 7885 + 68:5 + 8954 + + 7.51 
5.808 + .25 + 567.8 + 5 4896 + 49.795 + 8.3, 
7.08 + 23.04 + 8:2; + 848 + 3ole + rh, + 7.00019 
8.999, + 84 + 507 + 28 49, + =% + 6.8819 + 3.1416 


DECIMALS. 293 


SUBTRACTION OF DECIMALS. 


689. Give answers in decimals: 


275.8 — 8125 
38738 — 99.0127 
108/00) a 

. 62.365 — 483 
1982 — 13.6431 


BY. (hip 24 ve 9, 88 
58. 2,845 — 34545 
59. 168.3, — 54.8759 
60. 18.42—.576} 
61. 1,84714 — 344,9, 


MULTIPLICATION OF DECIMALS. 


Give answers in decimals: 
. 24.75 x 34 
982 x .00046 
. 1482, x 12.5 
3801 x .012 
wx X 1.48 


67. 19.5 x .000484 
68. 1.876 x 33 

69. 3.48 x 4.8665 
TO easel 

71. 192.38 x .288 


DIVISTON OF DECIMALS. 


691. Give 3 places of decimals in quotient, exclusive of ciphers. 
Divide without writing products (Arts. 385, 616): 


72. 


7.3845 + .29 

. 840,753 + 4.18 
4.054 + 18.25 

123.5 + 884 
AT] + 5.825 
8126 + .0134 
12.845 = .0047 
8756 + 4.3822 

8 + 122 
. 15.3678 + .9125 


82. 


83 


84. 
85. 
86. 
87. 
88. 
89. 
90. 
91. 


48.45 + .089 
. 89562.478 + 4279 
346.25 + 64.8 
9.1342 + 208.3 
1784 + 29.57 
843.71 + 1.127 
83.087 + 5.37 
137.84 - 7.91 
38.9008 + .523 
81074 + .009157 


294 ARITHMETIC, 


692. Solve by short division. 


When ciphers are canceled in the divisor, what change must 
be made in the decimal point of the dividend ? 


92. 18.756 = 300 102. 48.64 200 
93. 48.36 + 4,000 103. .00531 + 90,000 
94. .4824 + 12,000 104. 96.51 + 60 
95. 11.011 + 700 105. 87.5 + 500 
96. 3.6504 + 90 106. 183.275 + 10,000° 
97. 45.63 + 1,500 107. 1.7632 + 1,600 
98. 130.13 + 1,100 108. 1.5639 + 130 
99. .8712+ 60 109. 6144+ 120 
100. 3.075 + 5,000 110. .976 + 800 
101. .07056 = 140 111.  .8008 + 7,000 
MISCELLANEOUS. 


694. Slate Exercises. 


1. Find the cost of 24,400 bricks @ $6.25 per M. 


Ans. $6.25 x 24.400 = $6.25 x 24.4. 
(How do we divide by 1,000?) 


2. 760 pounds of hay @ 95 cents per cwt. (100 lb.). 
48,600 laths @ $2.80 per M. | 
39,250 stamped envelopes @ $21.30 per thousand. 
1,875 pounds of straw @ 68 cents per cwt. 
108,745 Philadelphia bricks @ $22.00 per M. 
14,860 oranges @ 75¥¢ per 100. 

2,576 eges @ 181 per doz. 

4,500 cigars @ $35 per M. 

10. 28 doz. wax candles @ $13.50 per gross (144). 


i i 


DECIMALS. 


295 


695. Solve by cancellation where possible : 
38,648 lb. of wheat @ 90 per bu. (60 lb.). 


11. 
12. 
13. 
14. 
15. 
16. 


93.8). 


696. Perform indicated operations. 


18,964 Ib. 
48,576 Ib. 
69,104 Ib. 
74,816 lb. 


of coal @ $5 per ton (2,000 lb.). 
of oats @ 36f per bu. (82 lb.). 
of rye @ 911¢ per bu. (56 lb.). 
of corn @ 481 per bu. (56 |b.). 


360 meters of cloth @ $1.10 per yd. (1 meter = 39.37 
inches). 


17. Cost in United States money of 386 hats @ 24 francs 
each (1 franc = 19.3f). 


18. 480 meters of cloth @ 1.10 marks per meter (1 mark = 


Change divisor to whole number, making corresponding change 
in the dividend. Cancel. 


19. 


21. 
22. 
23. 
24. 


25. 


7 
34.2 X ofp ae 
35 DET 239.4 


.249 x 3.92 
.098 


soe 12 
288 


6876 x .27 
081 


ele OO 
19.3 


3.1416 x 2.3 
£7804 


28. 


29. 


30. 


i) 


Hho 6 x 188 
OD X. he 
T2op 234 


450 x 23.8 
1.19 


p40 x Gul 


49 x 100 


576 X 6.3 
14.4 x 25 


TD OU L 
ese G 


306 x 8.75 
.9 X 68 


296 


ARITHMETIC. 


697. Reduce to common fractions — lowest terms: 


31. 
32. . 
2 
34. . 


43. 
44. 
45. 
46. 


35. 
36. 
37. 
38. 


47. 
48. 
49. 
50. 


699. Blackboard Exercises. 


Write answers at sight - 


1. 


ro ee ee ee 
ao fF WO wo KF OS 


OMDMWAT Pw D 


244 -+ 153 
132 x 6 
424 — 131 
81h 42 
502 + 204 
801 — 401 
5i x 5d 
242+ 2 
3864- 3 
17i+ 4 
. 214+ 5 
. 488-— 6 
. 182+ 7 
. 244+ 8 
. 362+ 9 


0064 
ou 
062 
8334 


16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 


39. .009 
40. .044 
41. .7612 
42. 0374 


bl. ae. 
52. ah 5. 
pS uae 
54. 35 
544 — 392 
488 — 4 
621 + 232 
123 x 6 
662 + 334. 
334 — 162 
13x 1% 
80 + $$ 
72 + & 
56 + 
80 
75 + 1t 
90 + 12 
98 + 13 
81. + 1% 


RECTANGLES. 297 


MEASUREMENTS. 


700. How many square inches in each of the following rect- 
angles? First change each dimension to inches. 


1. 42 in. by 36 in. 6. 9 ft. by 11 ft. 

2.) (Linpy 18 in. 7. 27 in. by 30 in. 

3. 3 ft. 1 in. by 4 ft. 2.in. 8. 65 in. by 92 in. 

4. 5 ft. 3 in. by 6 ft. 4 in. 9.071 tt. din. by: Arya. 
5. 12 ft. by 18 ft. 10. 3 yd. by 6 ft. 6 in. 


7OL. How many square feet in each of the following rect- 
angles? First change each dimension to feet, or to feet and 
a fraction. 


11. 18 ft. by 24 ft. 16. 3 ft. by Li yd. 

12. 36 in. by 4 ft. 17. 42 in. by 4 ft. 

13. 6 yd. by 8 yd. 18. 25 ft. by 17 ft. 6 in. 
14. 1 yd. by 48 in. 19. 42 in. by 48 in. 

15. 32 ft. by 4 ft. 20. 13 yd. by 15 yd. 


702. How many square yards in each of the following rect- 
angles? Change each dimension to yards, or to yards and a 
fraction. 


21. 18 yd. by 25 yd. 26. 36 yd. by 24 in. 
22. 15 yd. by 1 yd. 1 ft. 27. 17 ft. 6 in. by 32 in. 
23. 27 ft. by 36 ft. 28. 22 ft. 9in. by 18 in. 
24. 54 ft. by 2 ft. 6 in. 29. 108 in. by 90 in. 


25. 24 yd. by 27 in. 30. 180 ft. by 54 in. 


298 


ARITHMETIC. 


SPECIAL DRILLS. 


703. Give sums: 


185+ 89 450 + 690 
56 + 256 680 + 350 
394+ 77 870 + 260 
39 + 461 940 + 480 
704. Give differences : 
224 — 185 1,089 — 274 
3831 — 286 1,197 — 786 
442 — 369 1,258 — 680 
554 — 487 1,476 — 828 


705. Give products : 


82 x 25 
25 x 73 
64 x 25 
25 x 55 


98 x 4 
89 x 5 
78 x 6 
67x 7 


706. Give quotients : 


792— 9 
415+ 5 
ei Bpeomal 
406+ 7 


192 + 88 
380 + 76 
693 + 63 
486 = 54 


707. Give answers: 


123 x 5 
112 x 4 
103 x 3 
93 X 2 


3X 88 
4x 7§ 
DX 62 
6 x 5§ 


576+ 76 
85 + 646 
768+ 48 
56 + 575 


1,200 — 610 
1,460 — 780 
1,820 — 390 
1,210 — 240 


46 X 334 
39 X 383} 
26 X 333 
19 X 833 


975 = 25 
850 + 25 


675 + 25 | 


825 -~ 25 


45,x 7 
333 x 8 
2755 X 9 
1, x 10 


274 + 815 
783 + 306 
459 + 740 
624 + 585 


458 — 69 
315 — 87 
672 — 95 
818 — 29 


63 x 12 
54 x 11 
15 Keo 
36 no 


300 + 33} 
4331 + 334 
666% + 332 
5881 + 331 


11 x 2,5 
12 x 33 


11 x 43 
10 x 5% 


ee a a 


REVIEW. 299 


708. Oral Problems. 
1. I sold 875 bushels of wheat to one miller and 87 to 
another. How many bushels did I sell? 


2. Bought goods to the amount of $4.29. How much change 
from a $5 bill? 


3. What will be the cost of 89 tons of coal at $5 per ton? 
4. If 49 hats cost $147, what is the cost of one hat? 


5. 567 marbles are divided among 9 boys. How many does 
each receive ? 


6. How many yards in 5 pieces of cloth, each containing 
12% yd.? 
7. Divide 292 by 7. 
8. What will be the cost of a barrel of flour at $5.25 and 8 
lb. of sugar at 6 #? 
9. When silk is 75¢ per yd., how many yards can be bought 
for $9.75? 
10. If 23 yd. ribbon cost 42 cents, what will 33 yd. cost? 
11. How much must be paid for 55 lb. of raisins, at 8¢ 
per lb.? 
12. Find the cost of 320 lb. of hay at 60 per hundred pounds. 
13. If eggs are sold at the rate of 18 for 25 cents, what will 
be the cost of 6 dozen eggs? 


14. Three men require 22 days to do a certain piece of work. 
How long would it take 11 men to do the same work? 


15. A father earned $14.60, his son earned $7.80. What 
were the earnings of both? 

16. How many yards of fence will be required to enclose a 
rectangular field 98 yards long and 50 yards wide? 


17. A farmer divides his farm of 425 acres into fields of 123 
acres each. How many fields has he? 


300 ARITHMETIC. 


18. There are 36 inches in a yard. How many yards are 
there in 324 inches? 


19. The product is 925, the multiplier is 25. What is the 
multiplicand ? 

20. What will be the cost of 46 tons of hay, at $123 per ton? 

21. What is the weight of 25 firkins of butter, each contain- 
ing 56 pounds? | 

22. At $1.75 per yard, how many yards of cloth can be 
bought for $49? 

23. What price was paid for 20 sheep, at $8.75 per head? 

24. A man saved $320 per year for 5 years. How much 
more would he require to make $ 2,000 ? 


25. Mr. Jones sold a lot for $675, thereby losing $85. What 
did he pay for it? 


709. Slate Problems, 


1. The width of a room is 2 of its length. How many square 
feet in the floor, if the width is 15 ft. ? 


2. If 2 lb. 6 oz. of tea cost 95 cents, how 
many pounds and ounces can be bought for 
$ 2.35 ? 

3. What will be the duty on 175 kilograms of wool at 33 ct. 
per lb.? (1 kilogram = 2.2046 Ib.) 


4. John and James went out together. John had 38 cents. 
When one of the boys had spent 18 cents and the other had spent 
16 cents, they had 24 cents left between them. Find the amount 
of money James had. 


Find 4 of the sum of 2 and 3. 
What is 2 of the difference between ¢ and 3? 
What fraction added to ? gives }? 


Change 1,3, hour to seconds. 


15 ft. 


Sees cee 


REVIEW. 301 


9. £ of what number equals 180? 
10. The half of a number added to its fourth 213 
part equals 213. What is the number ? 4 4 
11. A farm is sold for $5,700, at a loss of 5 of the cost. 
What was the cost? , 


12. When it is noon at Philadelphia, it is 15 seconds and 
10 minutes past 5 p.m. at Paris. What time is it at Philadelphia 
when it is noon at Paris? 


13. A, B, and C buy a house. A furnished + of the cost, 


B 4, and © $1,200. What did A A poe ES A 
and B pay, respectively ? 4 + $1200. 


14. A room is 221 ft. long and 18 ft. wide. What will it 
cost, at.5¢ per yard, for a strip of moulding around the walls? 


15. How many square yards of carpet would be needed for 
the floor of the above room ? 


16. How much is the fraction $ increased or diminished when 
2 is added to each of its terms (numerator and denominator)? 


17. After James has spent # of his money and + of the re- 
mainder he has but $1.50 left. How much had he at first ? 


18. A man buys oranges at $1.20 per 100. How many 
would he have to sell, at 25% per dozen to gain $3.18? 


19. From a piece of cloth measuring 281 yards, there have 
been sold 22 yd., 68 yd., 18% yd. If the remainder is worth 
$13.10, what is the value of the whole piece? 


20. A man left for charitable purposes $3,600, which was 2 of 
his money. The remainder was divided equally among 8 rela- 
tives. How much did each relative receive? 


$ 3600. $2 


$ 
Charitable purposes Eight relatives 


302 ARITHMETIC, 


REVIEW. 


710. Supply missing numbers : 


1. $18,432.65 2. $26,459.88 3. $93,259.80 

9,876.04 6,087.90 10,059.77 

632.95 12,364.58 5,387.04 

27.88 3,030.30 20 

5.63 999.99 23.50 

99 6,875.84 681.19 

04 365.93 32,065.88 

87 ? 793.20 

2.90 6.50 2,684.39 
83.15 25.19 ? 

700.07 308.12 15,909.75 

4,862.99 4,321.00 123.40 

? 87 6.15 

$50,000.00 $ 76,543.21 $ 202,020.20 


711. Complete the following table of public school attendance: 


CoLORED. WHITE, 


x __|| Aggre- 

Male. | Female. | Total. || Male. | Female. | Total. ore 

AUDULN IS tween he 40 35 75 || 1,706| 1,753 | 3,459 || 3,534 
Binghamton . . 21 20 2320 Ieee i 
Brooklyn... 839 797 54,647 | 54,439 
Cohoes . . . .| — — a LSG2y Vigo 
HATAITA Rete int 40 61 23174 2211 
New York. . . 806 806 98,029 | 98,304 
Rochester . . . 31 38 8,258} 8,597 
PLOT Cuter ies, hy, 5 5 1,094 992 
Saratoga Springs. | .35 34 1,088) 1,116 


fy OUKOTS aasvuee de 15 15 1378) L718 


REVIEW. 


SHORT METHODS. 


713. Sight Exercises. 


. 68 x 25 
neo x 49 


e244 x 15 
- 82x 123 


o FF WW WO 


. 88x 12} - 


64 fa C25 Ly 
Meo OL 12. 
8. 66 X 334 13. 
9. 48 x 75 14. 


10. 24x62) 16. 


714. Slate Exercises. 


9,347 x 25 
863 x 75 
8,123 x 124 
6,483 x 334 
8,128 x 125 
9,847 x: 250 
9,347 x 22 
9,347 x 75 
6,483 x 662 
6,488 x 374 


715. Oral Problems. 
1. What will be the cost of 49 Ib. of coffee at 25 ¢ per lb. ? 
2. I paid $14.75 for eggs at 25% per doz. How many dozen 


did I buy? 


96 xX 25 
25 X 81 
48 x 374 
92 x 50 
82 X 334 


16. 
17. 
18. 
19. 
20. 


303 


88 x 25 
25 X 97 
16 x 874 
66 x 662 
16 x 663 


it, 
12; 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


4,896 x 871 
1,284 x 622 


75 xX 2,468 
334 X 3,870 
662 x 3,456 
162 X 1,266 


8,408 x 62} 
8,875 x 87h 
1,995 x 123 
7,314 x 250 


3. What will be paid for 104 bu. of wheat at 8719 per 


bushel ? 


4. How many bushels of corn at 624% per bu. can be bought 


for $150? 


304 ARITHMETIC. 


5. How much will be paid for 99 yd. of ee goods at 831¢ 
per yd.? 


6. How many yards of carpet at 662% per yard can be 
bought for $ 84? 


7. Find the cost of 15 doz. collars at 121 ¢ each? 


8. Paid $24 for cuffs at 162% per pair. How many dozen 
pairs were bought? 


9. What will be the cost of 128 lb. of tea at 75¢ per lb. ? 


10. A bale of cotton at 61f per lb. cost $25. What was the 
weight of the cotton ? 


11. A farmer sold hay at 75 per cwt., receiving for it $39. 
How many cwt. did he sell? 


12. How many bbl. of mess pork at $12.50 per bbl. can be 
bought for $175? 


13. What will be the cost of 96 yd. of carpet at $1.25 
per yd.? 


14. When wheat sells at $1.124 per bu., how many bushels 
can be bought for $198? 


15. At $3.50 each, what will be paid for 84 coats? 
16. Find the cost of 28 hats at $2.75 each? 


17. A real estate agent sold 97 lots at $250 each. How 
much did he receive for them? 


18. What will be the cost of 248 horses at $125 each? 


19. At 4 cent each, how many pen-holders can I buy for 
$ 5.76? 


20. Paid $3,675 for cows at $75 each. How many were 
bought? 


APPROXIMATIONS. ; 805 


716. Slate Exercises. 


Multiply 1875 by 21. Do not place the multiplier under- 
3750 neath the multiplicand. 
39,375 Ans. 
1. 3,456 x 31 Gl Goyee LOL 
2. 7,465 x 81 Zab hOs hao OL 
3. 2,345 x 41 $11,689) xP 
4, 5,482 x 91 9.1) 4,892 x 71 
5. 9,284 x 51 10. 10,754 x 121 
717. 2468 x 18. Write the product by 8 one place to the 
19744 right (above or below). 
44 424 Ans. 
11. 8,734 x 13 15. 57614 x 21 
12. 4,075 x 18 16. 345x15 X41 
13. 9,485 x 14 Tie 40 x Sica L 
14. 5,832 x 19 18 20x Ge OL 
APPROXIMATIONS. 


719. Give approximate answers, at sight (Art. 521): 
1. 232 lb. of tea @ 50L¢. 

24 horses @ $124.95. 

64 yd. of carpet @ 8733, ¢. 

485 bu. of wheat @ 993. 

96 lb. of coffee @ 247. 

840 yd. of dress goods (@ 33 

360 yd. of oil cloth @ 663£. 

48 owt. of straw @ 623%. 

92 hats @ $1.49}. 

128 lb. of lard @ 123%, 


g, 


5 
16 


OM AHA Pe ww 


_ 
sf 


306 


Inair © ND 


ARITHMETIC. 


720. Give approximate answers in whole numbers: 


11. $27 + 2413 21 
12. $299.96 + $1.492 22. 
13. $24.05 + 375% 23 
14. $15.03 + 128¢ 24 
15. $60+$ 2.4918 25 
16. $32 + 333¢ 26 
17. $69.95 + 871¢ 27 
18. $60+ 62,1.¢ 28 
19. $64 + 6611 29 
20. $27.95 + $1.75 30 
721. Sight Exercises. 
Give products : 
360 x .25 8. 840x .075 
560 x .125 955) 960. 9.005 
240 x .875 10. 1,200 x .001 
400 x .625 11. 1,500 x .002 
480 x .75 12: 96 x .84 
820 x .875 13. 840 x .022 
720 x .025 14. 1,500 x .06 
722. Give quotients : 
240 + .5 8. 87+ .05 
860 + .75 - 48+ .005 
AB + 195 10. 72+ .025 
23 4.25 11. 92+ .002 
360 + .875 12. 93+ .034 
100 + .625 13. 54+ .021 
154 + 875 14. 182+ .06 


- 17.3 x 3.98765 


256.008 x .249875 


15. 
16. 
17. 
18. 
19. 
20. 
21. 


15. 
16. 
ty 
18. 
19. 
20. 
21. 


» 20.1234 X 15.98 
aoa Wipe alae by 

. 86.4 x .996 

. 33.833 x 5.004 
. 799.387 x .125 
eo eos 

. 7.33 X 11.0083 
. 64.002 x .38750 


400 x .04 
165 x .062 
176 x .064 
3,300 x .004 
880 x .124 
105 x .8 
210 x .10 


76 + .04 
88 = .004 
65 + .123 
84 + .8 
11 + .064 
42 -+ .62 
93 + .5 


723. Give results: 


DENOMINATE NUMBERS. 


elite 6. Ay 75 12. 95x14 
@ He 13. 18 x 2§ 

74 X 24 
2S aera 7. © 49 14. 16 x 34 
ie 15. 14x 54 
3, 63x19 nano haben 
a 11 V7 Oe 
96 x 27 . 22+ 28 
fe BD See A 1S sores 
23 10. 72x14 19. 60+ 2} 
Be nex oo Ba 
8 11. 38 x 1} 20. 40+ 62 


DENOMINATE NUMBERS. 


724. Oral Problems. 


1. What will be the weight of 16 hams that average 10 lb. 
5 oz. each ? 


2. From a chest of tea containing 54 lb. there were sold 27 lb. 
70z. How many lb. remain? 


3. Seven bushels of potatoes are divided among 8 persons. 
How many pecks and quarts does each receive ? 


4. How many square inches in the surface of a sheet of paper 
measuring 11 inches by 13 inches? 


5. How many feet and inches in $ yd.? 
6. What decimal of a pound is 14 oz. ? 


7. A man buys a bushel of hickory nuts. After he sells 
2 pk. 4 qt., what fraction of the bushel has he left? 


8. A dealer puts 80 gal. of milk in cans holding 1 qt. 1 pt. 
each. How many cans does he fill? 


9. At $20 per month, how much rent will a man pay in 
1 year and 5 months? 


10. 75 hundredths of a pound is how many ounces ? 


308 ARITHMETIC, 


11. How many feet in 5 rods? 


12. 1 gal. 3 qt. 1 pt. of milk is divided among 5 people. How 
many quarts and pints does each receive? 


13. What fraction of 2 lb. 3 oz. is 1 lb. 4 oz.? 
14. Three-eighths of a ton is how many pounds? 
15. Change 9 hours 36 minutes to the fraction of a day. 


725. Slate Problems. 
1. 382 hams weigh 458 pounds. What is the average weight? 


14 lb. 5 oz. 
32458 lb. 
138 
10 lb. remainder. 
16_ 
160 oz., new dividend. 
0 Ans. 14 lb. 5 oz., average. 


2. 595 gal. of oil are put into 14 barrels. How many gal. 
and qt. does each contain? 


3. If there are 42 gal. and 2 qt. in a barrel of oil, how 
much oil will there be in 15 barrels? 


4. In the written number 54,372, the value expressed by the 
5 is how many times the value expressed by the 2? 


5. A piece of cloth containing 57 yd. is divided equally 
among six persons. What is the length of each one’s share? 


6. How many minutes in 1 day, 1 hour, and 1 minute? 


7. July 1 is the last school day. How many days’ vacation 
will there be, if school begins Sept. 6? 


8. How many hours and minutes are there from half-past 3 
Saturday afternoon to a quarter before 9 Monday morning? 


9. How many steps, 2 ft. 6 in. long, must a man take in 
walking 1,200 yards? 


FRACTIONS. 309 


10. A man owns a plot of ground 420 ft. long, 240 ft. wide. 
How many rods of fence will be required to enclose it ? 


11. A train goes from Jersey City to Washington, 228 miles, 
in 4 hours 12 minutes. How many miles an hour does it travel? 
How long does it take the train to go one mile? 


12. On Monday a boarding-house uses 3 gal. 2 gt. of milk; 
on Tuesday, 4 gal.; on Wednesday, 3 gal. 3 qt. 1 pt.; on Thurs- 
day, 4 gal. 2 qt.; on Friday, 6 gal.; on Saturday, 5 gal. 2 qt. 
1 pt.; on Sunday, 3 gal. 1 pt. How much is used during the 
week, and what is the average per day? 


13. June 21 the sun rises at New York at 4.23 a.m. and sets 
at 7.40 p.m. How long is the night? 


14. From 3% bu. take 37 pk. 


15. What is the length in rods of a fence surrounding a field 
206 ft. 3 in. wide and twice as long? 


REVIEW FRACTIONS. 


726. Slate Exercises. 


Add: 
293 + 17+ 62,5 


Weigelebaa. geal ay, 5 

432 + 30+ 632+ 8144 

114, + 258+ 428 + 82+ 91 
83.8, + 914 + 7018 + 6% +37 
244801474 6914 93814 
14+ 214 + 821 + 484 4 542 
927. + 68.5, + 33 + 724.89 
75519 + 951 + 303 + 524 13,5 
60x45 + 493% + 183% + 65 + 90} 


OM AFT Pw Do 


310 ARITHMETIC, 
727. Subtract: 
11. 481 — 258 16. 
12. 942 —184 17. 
13. 6954, — 3313 18. 
14. 577, — 2617 19. 
15. 100.3, — 5043 20. 
728. Multiply : 
21. 481 x 2 26. 
22. 82x 411 27. 
23. 168 x 127 28. 
24. 4,3 x 122 29. 
25. 24 X 24 X 2h 30. 
729. Divide: 
31. 75-2 36. 
32. 7+ 5 37. 
33. 188.7. +5 38. 
34. 1718-44 39. 
35. 183 -- 24 40. 


730. Perform indicated operations 


41. 


A2. 


43. 


44. 


(15+ 7) + (68 + 4) 
(} x 20) — (4g X 23) 


eS 
16 BL 
2of 4 lot lt 


234 + (84+ 18) 


45. 


46. 


47. 


48. 


126.8, — 832 
99,8, — 61% 
84,7, — 15%, 
23g5 — O75 
91219 — 68428 


$x 12x 354 
231 x 102 

88 x 93 

vo X 14 x 34 
162 X 8 X 834 


73 + 3,3, 
373 +15 
128.4, + 25 
5 -+- 94 

7 + 123 


524 x (1d — He 
7m GES uae tite 
9) 

SM a a 
LB ics 8 


7 of (3$ — 25 + 95) 


REDUCTION. Se 


DENOMINATE NUMBERS. 
732. Slate Exercises. 


Change : 

1. 48 pounds and 9 ounces to ounces. 

2. 34 rods and 3 yards to yards. 

3. 2 miles to yards. 

4. 3 days and 17 hours to hours. 

5. 24 minutes and 15 seconds to seconds. 

6. 8 tons and 1675 pounds to pounds. 

7. 43 gallons and 8 quarts to quarts. 

8. 75 gallons to pints. 

9. 19 bushels and 3 pecks to pecks. 
10. 19 bushels and 3 pecks to quarts. 
11. =, ton to pounds and ounces. 

12. .03125 ton to pounds and ounces. 


13. Z yard to feet and inches. 


735. Slate Exercises. 

Change: 

975 ounces to pounds and ounces. 
396 inches to yards. 

517 hours to days and hours. 

1,694 seconds to minutes and seconds. 
9,314 pounds to tons and pounds. 
987 pints to gallons, quarts, and pints. 
1,485 quarts to pecks and quarts. 

185 pecks to bushels and pecks. 

840 hours to weeks. 


312 


739. 


ARITHMETIC. 


DENOMINATE NUMBERS. 


. Slate Exercises. 


13 lb. 6 oz. 
5lb. 9 oz. 
25 |b. 10 oz. 


19 yd. 1 ft. 
2 ft. 
& yd. 1 ft. 


8 hr. 40 min. 
25 min. 
5 hr. 9 min. 


5 min. 80 sec. 
11 min. 25 sec. 


9 min. 18 sec. 


Leveson. 
2 ft. 6 in. 
Aisle tt. 4 ins 


Subtract : 


Ait ot 


4 lb. 7 oz. 


15 yd. 1 ft. 
9 yd. 2 ft. 


17 hr. 
9 hr. 50 min. 


40 min. 30 sec. 
6 min. 45 sec. 


1 yd. 1 ft. 1 in. 
2 ft. 9 in. 


10. 


10. 


18 gal. 3 qt. 
9 gal. 1 qt. 
2 qt. 

11 bu. 3 pk. 
6 bu. 2 pk. 

2 pk. 

1 pk. 6 qt. 

1 pk. 7 qt. 

5 qt. 


?) Sowks5 da. 


6 wk. 6 da. 
l wk. 3 da. 


NG Bd A a Relay at ay: 
4T. 983 1b. 
1756 |b. 


. 20 gal. 1 qt. 


6 gal. 3 qt. 
89 bu. 2 pk. 
67 bu. 3 pk. 
3 pk. 2 qt. 
2 pk. 7 qt. 
11 wk. 1 da. 


9 wk. 5 da. 


5 T. 896 Ib. 
1984 lb. 


REVIEW. Fad Bs: 


CANCELLATION. 
740. Slate Problems. . 
Indicate operations, and cancel where possible : 


1. If 56 men can pave a street in 24 days, how long will it 
take 32 men to pave it? 


2. When a vessel sails 168 miles a day, she completes her 
voyage in 14 days. In what time would she complete it if she 
sailed 196 miles a day ? 


3. If a field would support 64 sheep for 21 days, how long 
would it support 48 sheep ? 


4. If 42 men could build a wall in 24 days, how many men 
could build it in 18 days? 


5. If 21 horses are worth as much as 35 cows, how many 
horses are worth as much as 55 cows? 


6. A girl that wrote 36 letters to a line, took 15 lines in 
writing a piece of dictation. How many lines would a girl that 
wrote 30 letters to a line, require for the same dictation ? 

7. If a boy that steps 27 inches at a time takes 1,000 steps 
in going home from school, how many steps will be taken by a 
boy that steps 30 inches? 

8. If 1,920 bricks will build a wall 15 yards long, how many 
bricks will be required for a similar wall 24 yards long? 

9. A train going 44 miles an hour, went a certain distance in 
9 hours. How long would a train take that went 36 miles an 
hour ? 


10. Find the cost of one-eighth of a barrel of flour (196 lb.), 
at the rate of 11 cents for 34 pounds. 


11. Six men can do a certain piece of work in eighteen days. 
How long would it take eighteen boys to do the same work, if 
“one man can do as much work as two boys? 


314 ARITHMETIC. 


12. Ifa certain quantity of flour will last 48 persons 57 days, 
how long will it last 88 persons? 


13. Divide 2 of 56 by 12 times 32. 


ee 


DENOMINATE NUMBERS. 


742. Slate Exercises. 
Multiply : 


1. 12 1b. 7 02.x8 6. 4yd.1ft.x5 
2. 3 hr. 10 min. x 7 7. 7 min. 18 sec. x 10 | 
3. 4. 985 Ib. x 11 8. 9 gal. 3 qt. x 2 | 
4. 7 bu. 8 pk. x9 9. 2 ft. 9in. x8 | 
5. 8wk.4da.x4 10. lyd.1 ft. 6 in. x6 | 
743, Divide: 
11. 91b. 20z.+2 16. 18 yd. 2 ft. +7 
12. 31 gal. 2 qt.+9 17. 19 ft. 2 in. +10 
13. 19 hr. 21 min. +3 18. 34 T. 936 lb. +4 
14. 26 bu. 1 pk.+5 19. 17 wk. 1 da.=6 
15. 41 min. 44 sec. + 8 20. 52 yd. O ft. 9 in. + 11 
744, Divide: 


21. 18 lb. 4 oz. by 4 lb. 9 oz. 

22. 16 yd. by 2 yd. 2 ft. 

23. 2 da. 3 hr. 36 min. by 6 hr. 27 min. 
24. 47 min. 42 sec. by 5 min. 18 sec. 
25. 84 yr. 7 mo. by 12 yr. 1 mo. 

26. 19 da. 3 hr. by 2 da. 3 hr. 

27. 3 mi. 40 rd. by 125 rd. 


MEASUREMENTS. 315 


28. 103 T. 808 lb. by 8 T. 1,234 Ib. 
29. 52 gal. 2 qt. by 3 gal. 2 qt. 

30. 68 bu. 1 pk. by 5 bu. 1 pk. 

31. 30 ft. 8 in. by 1 ft. 11 in. 

32. 52 yd. 9 in. by 4 yd. 2 ft. 3 in. 
33. 51 wk. 3 da. by 2 wk. 6 da. 


MEASUREMENTS. 


745. Slate Problems. 
Make a diagram in each case: 


1. A lot 25 ft. by 100 ft. has on it a house 25 ft. by 55 ft. 
How many square feet are there left for a yard? 


2. How many square feet are there in the floor of a room 
24 ft. long, 18 ft. wide? 


3. How many square yards are there in the ceiling of the 
same room ? 


4. Find the number of square yards of plastering needed 
for the end wall of a room 18 ft. wide, 9 ft. high, after deducting 
for two windows each 6 ft. high, 44 ft. wide. 


5. How many square yards of plastering will be needed for 
the opposite wall of the same room, 18 ft. wide, 9 ft. high, after 
deducting for a door 74 ft. high, 6 ft. wide? 


6. Calculate the number of square yards of plastering needed 
for two side walls of a room 24 feet long, 9 feet high, after 
deducting for a fireplace 6 feet square on one side. 


7. A house 30 ft. by 60 ft., with an addition 15 ft. square, 
is built upon a lot 100 ft. square. How many square feet of 
ground are covered by the building? How many square feet 
remain for a garden? 


316 ARITHMETIC. 


8. Measure the top of a brick and calculate the number of 
square inches in its surface. How many square inches in the 
surface of the bottom of the brick? Measure one side, and cal- 
culate its surface. How many square inches are there in the 
surface of the opposite side? How many square inches in each 
end? 

9. Measure a crayon box, and calculate the number of square 
inches in each face. 

10. Calculate the number of square feet in the floor of the 
class-room. In the ceiling. In each side wall. In each end 
wall. 


PERCENTAGE. 


746. Per cent means hundredths. Six per cent means six 
hundredths, ;§5, or .06. It is written 6%. 


747. Oral Exercises. 


1. What is ;8, of 200? 11. 4% of 125 
2. Find .06 of 300 12. 7% of 500 
3. 6 per cent of 400 13. 5% of 240 
4. 6% of 50 14. 1% ot 600 
5. 6% of 150 15. 49% of 600 
6. 6% of 250 16. 1% of 600 
7. 6% of 125 17. 24% of 600 
8. 6% of 75 18. 34% of 400 
9. 6% of 60 19. 4% of 400 
10. 6% of 160 20. 9% of 90 


748. What fraction equals: 


21. 25% 25. 20% 29. 62% 
22. 121% 26. 50% 30. 374% 
23. 331% 27. 61% 31. 624% 


24. 162% 28. 34% 32. 874% 


PERCENTAGE. 317 


749. Find: 
33. 50% of 96 42. 300% of 140 
34. 25% of 72 43. 150% of 140 
35. 121% of 120 44. 250% of 140 
36. 61% of 48 45. 125% of 140 
37. 334% of 36 46. 1% of 140 
38. 162% of 126 47. 1% of 350 
39. 841% of 72 48. 2% of 350 
40. 100% of 140 49. 39% of 350 
41. 200% of 140 50. 4% of 350 


750. Slate Problems. 


1. A house is valued at $24,500. How much taxes must 
the owner pay at the rate of $22.40 per $1,000 valuation ? 


2. A consignee sells a lot of cotton for $1,872.50. He 
receives 2% of this amount as commission. How much is his 
commission ? 


38. I loan $600 at 6% interest per year. How much interest 
should I receive from Jan. 1, 1892 to Jan. 1, 1894? 


4. How much will it cost me to insure goods to the amount 
of $18,760 at one per cent? 


5. A dealer imports books worth $548.40, on which he pays 
duty to the government at the rate of 25%. What is the amount 
of the duty ? 


6. Highty per cent of a class of 55 pupils are promoted. 
How many are not promoted? 


7. A man buys a house for $16,000 and sells it at a profit of 
3 per cent. How much does he gain? 


318 ARITHMETIC, 
8. A clerk spends for rent 18 per cent of his income of 
$1,850 per year. What rent does he pay? 


9. A girl spelled correctly 95 per cent of 60 words. How 
many did she miss? 


10. Tea costing 40 cents per pound is sold at a profit of 50 
per cent. What is the selling price? 


751. Oral Problems. 


1. What per cent does a boy receive if he solves 16 examples 
of the 20 given out? 


2. What is the interest on $200 at 4% for 2 years? 


3. If 2% yd. of calico cost 22 cents, how many yards can be 
bought for 60¢ ? 


4. What part of a ton is 125 pounds? 
5. How old, Dec. 1, 1892, was a boy born Sept. 1, 1879? 
6. What is the cost of 3,500 bricks at $20 per M? 


7. How many sheep, at $5 each, should be given in exchange 
for 12 horses, worth $200 each ? 


8. Reduce J, to a decimal. 


9. One hundred fifty marbles are divided among a certain 
number of boys. Each receives 12 and there are 6 remaining. 
How many boys are there? . 


10. At 3 oranges for 5 cents, what will be the cost of 4 dozen 
oranges ? 
11. 75 men can do a certain piece of work in 9 days. How 


long will it take 45 men to do the same work? 


12. If it takes 24 yards of carpet, a yard wide, to cover a 
floor, how many yards ? yd. wide will be needed for the same 
floor ? 


FEDERAL MONEY. 319 


BILLS. 


753. PHILADELPHIA, Sept. 24, 1894. 
Mr. Harrison JARVIS 
To Wu. Hart & Son, Dr. 


=e 


To 50 lb. of Pipe @, 51g 


“3 Faucets ‘ 75“ 
“1 Sink 4 | 75 
“ 33 days’ Labor ‘“ $4.75 


— 


1. Copy and complete the above bill. 


2. R. W. Jones has done 34 days’ work, @ $3.50 per day, 
for Charles Johnston. He charges for 850 ft. lumber at $2 per 
hundred; 5 lb. of nails at 9% per lb.; 8 locks @ 50¥; 2 bolts at 
10¢. Make out his bill. 


3. A gardener furnishes 3 rose-bushes at 75%; 4 grape-vines 
at 50%; 11 fuchsias at 830%; 25 pansies at 10%. He charges 
$3.25 per day for 23 days’ labor. Make out his bill. 


4. An upholsterer charges $3 per day for repairing some fur- 
niture. He supplies 6 lb. of hair at 50¢ per lb.; 17 yd. of 
plush at $1.75 per yd.; 3 papers of tacks at 10%; cord, gimp, 
etc., 47%. He works 4 days. Make out his bill. 


5. Make out and receipt a bill for four articles bought to-day 
by John R. Brown from Smith and Robinson, grocers. 


6. Make out a bill containing ten items bought by Mrs. S. W. 
Robb, at different times during October, 1894, from Frederick 
Loeser & Co., dealers in dry goods. 


7. Make out a bill for labor done and materials furnished by 
Anthony Jones, gardener. 


320 ARITHMETIC. 


INTEREST. 
754. Oral Exercises, 
What will be the interest on $100 for 1 year at 4% ? 
On $200 for a year at 5%? 
On $300 for a year at 6% ? 
On $400 for a year at 7% ? 
On $250 for a year at 4%? 


At 4% per year, what will be the interest : 
On $200 for 1 year? 

On $300 for 2 years? 

On $100 for 3 years? 

On $200 for 14 years? 

On $200 for 1 year 6 months? 

. What will be the interest on $200 for 3 years at 5%? 
- On $300 for 2 years at 6% ? 

. On $400 for 6 years at 3% ? 

. On $100 for 5 years at 7% ? 

. On $250 for 2 years at 4% ? 

. On $100 for 1 year 6 months at 6%? 
17. On $200 for 3 months at 4% ? 


At 4% per year, what will be the interest: 
18. On $200 for 6 months? 
19. On $3800 for 4 months? 
20. On $400 for 3 months? 
21. On $300 for 2 months? 
22. On $150 for 1 month? 
23. Find the interest on $ 24 for 1 year at 5%. 
24. On $36 for 1 year at 4%. 
25. On $67 for 1 year at 3%. 


a a 
Gi.2 OC Het 2635-19 et 


INTEREST. PAS 


755. Slate Exercises. 
Find the yearly interest on: 


1. $286.50 at 4% 11. $1,257 at 7% 

2. $485 at 6% 12. $168 at 33% 

3. $375.40 at 5% 13. $244 at 54% 

4. $379 at 3% 14. $890 at 73% 

5. $486 at 44% 15. $63.75 at 4% 

6. $186.75 at 4% 16. $937.50 at 6% 

7. $199.50 at 2% 17. $980.40 at 5% 

8. $636 at 34% 18. $159.60 at 24% 
9. $84.70 at 6% 19. $1,357.37 at 7% 

10. $93.25 at 8% 20. $2,146.18 at 44% 


Find the interest on: 


21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 
33. 


$ 290 for 2 years at 4% 

$ 1,400 for 3 years at 44% 

$ 2,840 for 4 years at 5% 

$ 1,250 at 6% for 3 years 

$5,360 at 54% for 2 years 

$380 at 3% for 44 years 

$ 780 for 1 year 4 months at 6% 

$ 2,560 for 2 years 6 months at 5% 
$ 1,025 for 3 years 3 months at 4% 
$ 1,296 for 7 months at 7% 

$ 648 for 5 months at 5% 

$275 for 4 months at 3% 

$1,000 for 11 months at 6% 


322 ARITHMETIC. 


AREAS OF RIGHT-ANGLED TRIANGLES. 


756. The square shown in the diagram is divided into two 
parts by a diagonal. One side of the square measures 10 feet. 


1. Mark in each triangle its area. 


Square. Rectangle. 


2. Divide a rectangle 20 ft. by 12 ft. into two parts by a 
diagonal. Mark in each triangle its area. 


3. Draw a right-angled triangle 3 in. by 4in. Calculate its 
area in square inches. 


4. How many square yards in the surface 
of a right-angled triangle whose base measures 
30 feet, and whose perpendicular measures 224 
feet. 


Perpendicular 


Find the area in square feet of the following right-angled 
triangles. (Change each dimension to feet.) 


Base 20 yd., perpendicular 380 ft. 
Base 16 in., perpendicular 3 ft. 
Base 30 in., perpendicular 1 yd. 
Base 3 ft. 6 in., perpendicular 5 ft. 


Re eg 00 ae = 03: - Ot 


Base 2 yd. 1 ft., perpendicular 1 yd. 9 in. 
10. Base 50 yd., perpendicular 36 yd. 


758. Multiply. Do not reduce to improper fractions. 


128 
x 45 
51 4 times 123 = 51; 3 of 123 = 41. 
uaa 
55L 
1. 182 x 64 6. 
2. 258 x 84 qs 
3. 164 x 54 8. 
4. 361 x 94 9. 
5. 223 x 6. 10. 
759. Divide. Short Division. 
11. 15) 752 _ 19 
12. 18) 169.8, 20 
13. 14) 2958 21 
14. 12) 9765. 22 
15. 16) 1953 23 
16. 21) 2314 24 
17. 26) 2634 25 
18. 24) 5044 26 


SHORT METHODS. 


SHORT METHODS. 


. Base 1123 ft., perpendicular 30 yd. 

. Base 90 in., perpendicular 2 ft. 

. Base 121 yd., perpendicular 134 yd. 

. Base 1 rod, perpendicular 74 ft. 

. Base 334 ft., perpendicular 18 ft. 6 in. 


163 x Tt 
483 x 121 
37% X 10% 
268 x 92 
323 X 87 


. 17) 1993 
. 18) 2214 
. 12) 8652 
. 18) 8924 
. 14) 5673 
. 28) 702% 
. 25) 8052 


. 22) 4653 


323 


324 ARITHMETIC. 


REVIEW. 


763. Answers in fractions: 


RO ti atans Lee pha 
Les (SUM Dlidy ste eed 
implity 34 1b oR 
2. Add 4, ye 20> a's) ao0- 
jee Re at txt 
3. Reduce 2 ' 8 4 to a simple fraction. 
144-4-3 
4. Divide (4+4+2) by ¢xHx #8) 
5. Simplify +3 of 3 of 4. 
1—i 14 
6. TRS ee en? 
1 


7. Find the value of q ers 


8. (244 14) + (24+ 33) =? 

9. Find the value of 24 times the quotient of (8 — 21) + 
(23 ie 3): 

10. 83¢4+14—7245—-4=? 


764. Answers in decimals : 

11. Divide the sum of .736 and 1.2854 by their difference. 

12. Divide .1 by .2 and .35 by 35, and find the product of the 
quotients. 

13. Reduce <74, to a decimal, and divide it by .3125. 

14. Divide .12096 by .082. 

15. Multiply .00273 by 3,000.456, and divide the product 
by .08. 

16. Divide 12.8125 by .000625. 


LY 
18. 


19. 


20. 


REVIEW. B20 


Divide 51.5 by 412, and 412 by 51.5. 
Multiply 31.5 by 27.9, and divide the product by 9.765. 


4.25 
844 


Reduce 


0021 x 8.004 


Find the value of 
024 


765. Reduce to lowest terms: 


1. of B. Ay 9. Hy 
2. Hf 6. sits 10. 3848 
8. $4 1. $48 11. 98h 
4. oH 8. Hig 12. $485 


CHAPTER IX. 


DENOMINATE NUMBERS. — SURFACES AND VOLUMES. — 
PERCENTAGE. — INTEREST. 


REDUCTION OF DENOMINATE NUMBERS. 


766. Reduce 5 gal. 3 qt. 1 pt. to pints, 
5 gal. 3 qt. 1 pt. 
In 5 gal. there are 20 qt. Adding the 3 qt, we _4 
have 23 qt. Multiplying by 2 to reduce to pints, 23 qt. 
and adding in 1 pt., we get the answer, 47 pints. 


47 pt. Ans. 

767. Reduction Descending, Slate Exercises, 
Reduce to pints: 

1. 16 gal. 1 qt. 1 pt. 6. 314 gal. 

2. 27 gal. 2 qt. 7. 9 gal. 23 qt. 

3. 16 gal. 8. 10 gal. 2 qt. 1 pt. 

4. 16 gal. 1 pt. 9. 27 gal. 1 pt. 

5. 34 gal. 3 qt. 1 pt. 10. 4 gal. 3 qt. 14 pt. 


768. Change 67 pt. to gallons, quarts, and pints. 
2| 67 pt. 
Ars qt. 1 pt. 
8 gal. 1 qt. 1 pt. Ans. 


769. Reduction Ascending. 
Change to gal., etc.: 


11. 156 qt. | 16. 277 pt. 
a ote 17. 139 pt. 
13. 408 pt. 18. 171 qt. 
14. 1,502 pt. 19. 63 qt. 

15. 63 pt. 20. 711 pt. 


326 


DENOMINATE NUMBERS. 


770. Reduction Descending. 


21. 
22. 
23. 
24. 
25. 
26. 
27: 
28. 
29. 
30. 
31. 
. 4wk. 6 da. 11 hr. to hours. 


. # of a week to hours. 


Change 17 yd. 1 ft. 9 in. to inches, 
4 mi. 100 rd. 4 yd. to yards. 
74 bu. 2 pk. 7 qt. to quarts. 


156 lb. 11 oz. to ounces. 
63 yd. O ft. 3 in. to inches. 


19 bu. 0 pk. 3 qt. to quarts. 


11 rd. 34 yd. to feet. 
63 gal. 3 qt. to pints. 
3 bu. 6 qt. to quarts. 


17 T. 369 lb. to pounds. 


15 hr. 16 min. to seconds. 


. wy of a mile to yards. 
. .00125 T. to ounces. 


771. Reduction Ascending. 


Change : 

36. 1,876 in. to yd., etc. 43. 
37. 475 oz. to lb., etc. 44. 
38. 729 qt. to bu., etc. 45. 
39. 8,675 min. to da., etc. 46. 
40. 4,972 lb. to T., ete. 47. 
41. 972 rd. to mi., etc. 48. 
42. 117 pt. to gal., etc. 49. 


9,483 sec. to hr., etc. 
877 qt. to bu., ete. 
1,495 oz. to lb., etc. 
373 in. to yd., etc. 
216 qt. to gal., ete. 
876 rd. to mi., etc. 
319 pt. to gal., etc. 


50. 3,520 yd. to mi. 


328 


ARITHMETIC. 


772. Oral Exercises. 


1G 


a 
Cont oar wn & © 


How many hours in 2 of a day? 


. How many hours in 4 of a day? 


How many minutes in 4 of an hour? 
How many hours and minutes in 4 of a day? 


How many quarts and pints in 3 of a gallon? 


. How many hours and minutes in .2 day? 


How many quarts and pints in .375 gallon? 
Change .3 day to hours and minutes. 
Change .625 bu. to pk. and qt. 

What part of a gallon is 1 pt.? 


. What part of a gallon is 3 pt. ? 

. What part of a gallon is 1 qt. 1 pt.? 

. What decimal of a gallon is 1 qt. 1 pt.? 
. What decimal of a gallon is 2 qt. 1 pt.? 
. What part of 2 gallons is 2 qt. 1 pt.? 

. Change .375 bu. to pt. and qt. 

. What decimal of a bu. is 4 qt.? 

. What fraction of a day is 3 hr. 20 min.? 
. Reduce 960 min. to hours. 

20. 


How many minutes in a day ? 


773. Slate Exercises. 


1. 


What decimal of a ton is 3 lb.? 


2. What fraction of a day is 12 min. 30 sec. ? 
3. 
4. Reduce .03125 day to minutes. 


Reduce 3), of a day to minutes. 


DENOMINATE NUMBERS. 329 


5. What decimal of a day is 9 minutes? 


6. What will be the cost of 15 T. and 750 lb. coal at $5 
per ton? 


7. If coal is $5 per ton, how much can be bought for $18.76 ? 


8. If7 T. 296 lb. coal cost $35.74, how much will I have to 
pay for 18,748 lb. ? 


9. A man pays $48.92 for 9 T. 1,568 lb. coal. How many 
tons, etc., would he receive for $ 73.11? 


10. Change 2 ft. 7 in. to the fraction of a yd. 
11. Reduce 3 pk. 4 qt. 1 pt. to the decimal of a bu. 
12. How many pk., qt., etc., in .9375 bu. ? 


13. If .1875 of a gal. of cologne cost $1.125, what will 1 pt. 
cost ? 


14. Find the cost of 42 gal. 3 qt. 1 pt. oil, at 16% per gal. 
15. Reduce 14 of a gal. to qt. and pt. 
16. What part of 3 T. is 1 T. 960 |b. ? 


. 17. A man raised 189 bu. 2 pk. and 2 qt. of rye. He sold 
119 bu. 2 pk. 4 qt. What fraction of his crop did he sell ? 


18. 10 bu. 1 pk. of seed is packed in 8 bags. How much is 
there in each bag? 


19. What decimal of a day is 21 hr. 14 min. 24 sec.? 


20. How many feet in a mile? 


COMPOUND ADDITION. 
774. Add the following : 


1. 17 \b. 3802. 2. 18 bu. 3 pk. 7 qt. 
4 lb. 9 oz. 9 bu. 2 pk. 4 qt. 
23 lb. 12 oz. 14 bu. 1 pk. 6 qt. 


15 oz. 2 pk. 


330 ARITHMETIC. 
Se LO Yara tt. 0): T.lL2 eros, 
17 yd. 4 in. 8 T. _980,1b. 
INSTA ee ba 476 lb. 
11 in. 1 T. 1,830 Ib. 
4.11 da. 5\hr, 19 min: 8. 2 wk. 5 da. 12 hr. 
23 da. 40 min. 6 da. 15 hr. 
17 hr. 50 min. 5 wk. 2 hr. 
5 da. 20 hr. 6 min. 2 da. 19 hr. 
5. 93 gal. 3 qt. 1 pt. 9. 18 mi. 100 rd. 
TA gal. 34 rd. 
18 gal. 1 qt. 29 mi. 
2 qt. 1 pt. 6 mi. 160 rd. 
6. 5 hr. 30 min. 20 sec. 10. 47 yr. 11 mo. 
45 min. 33 sec. D yr. 9 mo. 
6 hr. 11 min. 5 sec. 7 mo. 
10 hr. 38 min. 30 sec. 22 yr. 5 mo. 


11. 487 T., 316 T. 1,816 lb., 247 lb., 43 T. 811 1b.,19 T. 25 Ib. 

12.) 88 .1b}'15:0z,; 9. lb. 5.02; 18 Ib.) 22 Ibsi1L oz.; & Ib. 8 iog;, 
12 oz. 

13. 8 hr. 15 min. 5 sec., 37 min. 52 sec., 5 hr. 48 min., 23 hr. 
59 min. 5 sec. 

14. 72 gal. 3 qt. 1 pt., 17 gal. 1 qt., 2 qt. 1 pt., 90 gal. 1 pt. 

15. 7 yd. 2 ft. 11 in., 19 yd. 6in., 105 yd.,4 yd. 2 ft. 2in., 1 ft. 

16. 93 mi. 300 rd., 87 mi. 154 rd., 194 rd., 3 mi. 175 rd., 9 mi. 

17. 82 yr.1 mo., 19 yr. 10 mo., 25 yr. 9 mo. 6 da., 8 mo. 15 da. 

18. 4 wk. 6 da. 17 hr., 20 wk. 5 da., 4 da. 11 hr., 9 wk. 5 da. 
ST ah 

19. 5 hr. 13 min. 23 sec., 16 hr. 27 min. 30 sec., 48 min. 5 
sec., 24 sec. 

20. 8 bu. 3 pk. 7 qt., 5 qt., 2 pk. 1 qt., 4 bu. 6 gt., 3 bu. 1 pk. 
1 qt. 


Se ee 


DENOMINATE NUMBERS. Bal 


775. Find answers: 
UG WARE PIL oul itey A 
+ se 
VER Layo Pes: OZ. 
22. 14 bu. 2 pk. 4 qt. 
+ ? 
18 bu. 1 pk. 1 qt. 


ase 1 yde2 ft.7 in. 
AY 


26. 10 hr. 15 min. 80 sec. 
ae 


24 hr. 
OT ReAO TL LoLOribs 
an 
495° Teh od2 Ip, 


28. 9wk. 6 da. 11 hr. 
oe 


15 yd. 2 ft. 2 in. 21 wk. 3 hr. 
24. 19 da. 14 h. 40 min. 29. 84 mi. 24 rd. 
1" a 
30 da. 100 mi. 15 rd. 
25. 14 gal. 2 qt. 1 pt. 30. 13 yr. 9 mo. 
ots af 
18 gal. 1 qt. 20 yr. 
31. 83 lb. 4 oz. +? = 100 lb. 
32. 16 bu. 2 qt. + ? = 25 bu. 1 pt. 
33. 1 ft.4in.+?=9 yd. lin. 
34. 47 da. 15 min. + ? = 60 da. 


35. 


93 gal. 3 qt. 1 pt. +? = 150 gal. 


COMPOUND SUBTRACTION. 


776. Subtract: 
36. 83 yr. 3 mo. 
15 yr. 9 mo. 


37. 62mi. 84 rd. 
19 mu 159ird: 


38. 76'T. 295 Ib. 


Seek OF 2 lb, 


39. 100 lb. 


332 


40. 


41. 


42. 


46. 
47. 
48. 
49. 
50. 


ARITHMETIC. 
52 wk. : 43. 16 yd. 9 in. 
13 wk. 8 da. 7 hr. _T yd. 1 ft. 11 in. 
19 gal. 1 pt. 44. 100 bu. 
8 gal. 3 qt. 42 bu. 3 pk. 7 qt. 
18 hr. 5 min. 45. 45 da. 1 hr. 1 min. 
40 min. 25 sec. 6 da. 6 hr. 6 min. 


From 27 bu. 1 pk. 5 qt. take 18 bu. 8 pk. 7 qt. 

From 100 gal. 1 qt. take 83 gal. 2 qt. 1 pt. 

From 22 hr. 15 min. 20 sec. take 15 hr. 45 min. 40 sec. 
From 17 lb. 2 oz. take 18 lb. 8 oz. 

From 100 bu. take 74 bu. 2 pk. 1 qt. 


COMPOUND MULTIPLICATION. 


777. Add 3\b.9 oz. Add 4 gal. 3 qt. 1 pt. 


3 lb. 9 oz. 4 gal. 3 qt. 1 pt. 
4 gal. 3 qt. 1 pt. 


Multiply 3 lb. 9 oz. by 2. 
Multiply 4 gal. 3 qt. 1 pt. by 3. 


778. Multiply: 


. 13 bu. 3 pk. 6 gt. by 2. 59. 2 pk. 7 qt. by 10. 

. 20 gal. 2 qt. 1 pt. by 3. 60. 3 qt. 1 pt. by 11. 

- T lb. 10 oz. by 4. 61. 4 yr. 6 mo. by 12. 

. Shr. 15 min. 15 sec. by 5. 62. 5 wk. 6 da. 12 hr. by 16. 
. 23 bu. 8 qt. by 6. 63. 4 T. 250 Ib. by 18. 

. 82 gal. 1 pt. by 7. 64. 3 yd. 1 ft. 6 in. by 22. 

. 25 |b. 4 oz. by 8. 65. 2 mi. 15 rd. by 382. 


. 83 min. 38 sec. by 9. 66. 4 hr. 15 min, 20 sec. by 9. 


DENOMINATE NUMBERS. $3 


67. 31 gal. 2 qt. by 42. 71. 1 bu. 2 pk. 3 qt. by 13. 
68. 4 qt. by 37. 72. 4 yd. 2 ft. 9 in. by 15. 
69. 43 sec. by 215. 73. 21 hr. by 24. 


70. 4 wk. 6 da. 20hr. by 19. 74. 3 yr. 11 mo. by 14. 
75. 1 gal. 1 qt. 1 pt. by 30. 


COMPOUND DIVISION. 


779. Divide: 
76. 15 lb. 9 oz. by 3. 88. 13 wk. by 5. 
77. 2 1b. 3 oz. by 5. 89. 74 mi. 80 rd. by 4. 
78. 2 gal. 1 qt. by 3. 90. 69 yr. by 12. 
79. 5 bu. by 4. 91. 27 bu. 3 pk. 4 qt. by 2. 
80. 7 hr. by 6. 92. 76 gal. 3 qt. 1 pt. by 3. 
81. 17 lb. 7 oz. by 3. 93." hr. Liminv 7 sec. by 9: 
82. 37 bu. 3 pk. 6 qt. by 2. 94. 33 wk. 3 da. by 12. 
83. 67 yd. 2 ft. by 4. 95. 36 yd. 6 in. by 7. 
84. 33 da.15 hr. 57 min. by 3. 96. 45 bu. 6 qt. by 6. 
85. 563 gal. by 6. 97. 20 da. 13 hr. 4 min. by 16. 
86. 22 hr. 20 min. 20sec. by 4. 98. 15 gal. 3 qt. by 18. 
87. 112 T. 125 lb. by 5. 99. 54 yd. 1 ft. 4in. by 20. 


100. 41 wk. 4 da. 1 hr. + 4. 

101. 457 hr. 37 min. 30 sec. + 9. 
102. 147 gal. 3 qt. 1 pt. +18. 
103. 157 bu. 3 pk. 6 qt. + 7. 
104. 175 yd. 2 ft. 6 in. + 10. 
105. 188 mi. 12 rd. 2 yd. + 6. 
106. 311 da. 21 hr. 36 min. + 12, 


334 ARITHMETIC. 


107. Divide 180 da. 3 hr. 4 min. by 16. 


11 da. 6 hr. 11 min. 30 see. 
780. Dividing 180 days by 16, 16)180 da. 3 hr. 4 min, 


we get 11 days quotient and 4 20 
days remainder. Reducing 4 days 5) By (remainder) 
to hours and adding 3 hours, the 24 
next dividend is 99 hours. This — 
gives 6 hours quotient, 3 hours 99 hr. 
remainder. Reducing to minutes 3 hr. (remainder) 
and adding 4 minutes, the next 60 
dividend is 184 minutes. This 184 fats 
gives 11 minutes quotient, and 8 ——7 
minutes remainder. Reducing, we 24 
have 480 seconds for next dividend. 8 min. (remainder) 
Dividing, as before, the last quo- 60 
tient is 30 seconds. 480 Bec. 
0 


108. Divide 236 gal. 1 qt. by 18. 
109. 384 yd. 9 in. by 21. 


781. 15 yd. 2 ft. 9 in. 
.21)334 yd. 0 ft. 9 in. 
124 
19 yd. (remainder) 
3 
57 ft. 
15 ft. (remainder) 
Eb: 
189 in. 
0 
110. 825 lb. by 48. 116. 288 hr. 9 min. by 54. 
Lee Tby 25. 117. 863 gal. 2 qt. 1 pt. by 47. 
112. 483 mi. by 32. 118. 83 wk. 1 da. by 72. | 
113. 84 yr. by 24. 119. 1,188 T. 910 lb. by 81. | 
114. 462 bu. by 36. 120. 1,629 yd. 1 ft. by 96. 


115. 1,078 yd. by 63. 121, 1,867 gal. 13 pt. by 125. 


DENOMINATE NUMBERS. 335 
782. Oral Problems. 


1. How many tons and pounds of coal in 40 bags, each con- 
taining 80 pounds? 


2. If it takes 3 hr. 20 min. to hoe a row of corn, how long 
will it take to hoe 3 rows? 


3. A man puts up 34 pounds of tea into 4 oz. packages. 
How many packages does he make? 


4. 3 pecks 3 quarts of apples are divided among 9 children. 
What quantity does each child receive? 


5. What part of a day is 30 minutes? 


6. If there are 24 gallons of wine in 12 bottles, how many 
pints are there in each bottle? 


7. What is the weight of two packages each containing 
foitb, 11 oz, ? 
8. What part of an hour is 40 seconds? 
9. What is the rent of a house for 1 year 9 months at $16 
per month? 
10. If 3 gal. 2 qt. 1 pt. of milk are taken from a can contain- 
ing 10 gal., how much is left in the can? 
11. 5 hams weigh 614 1b. What is the average weight? 


12. There are on an average 41 pupils in a class. How many 
are there in 14 classes? 


13. At 371 cents per yard, how many yards can be bought 
for $6.75? 


14. Find the cost of 16 bbl. of flour at $ 6.124 each. 


15. $1.65 is equally divided among 15 boys. What is the 
share of each? 


16. A floor containing 404 square yards is 7 yards long. 
How many yards wide is it? 


17. How many ounces in 57 pounds? 


336 | ARITHMETIC, 


783. Slate Problems. 


1. If a watch gains 1 min. 17 sec. per day, how much will 
‘it gain during March and April? 

2. How many bu., pk., and qt. in 1,449 lb. corn, weighing 
56 lb. to the bu. ? 


3. A chain, 97 yd. 8 in. long, contains 1,000 links. Find 
the length of one of the links. 


4. A farmer sold out of 5 bu. of peas the following quan- 
tities: 3 pk. 6 qt.; 4 pk; 4 pk. 3 qt.; 1 bu. 1 pk. 1 qt. How 
much has he still to sell? 

5. A man walks on Monday 15 mi. 161 rd.; Tuesday, 10 m1. 
84 rd.; Wednesday, 19 mi. 15 rd.; Thursday and Friday, 12 mi. 
121 rd. each day; Saturday, 14 mi. 240 rd. What distance per 
day does he average? 

6. If the sun rises at 5 hr. 10 min. A.m., and sets at 6 hr. 
42 min. P.M., how long is the day? How many hours and minutes 
of night ? 

7. An iron rod is 12 ft. 6 in. long. From it are cut 73 bolts, 
each 13 in. long. How much is left? 

8. A man rows a mile in 10 min. 30 sec. How long would 
he take to row 27 miles at the same rate? | 

9. A man rows 51 miles in 23 hr. 5 min. and 30 sec. How 
long does he take to row a mile? 

10. If I lost $50 by selling a lot for two-thirds of its cost, 
what would I have lost if I had sold it for three-fourths of its 
cost ? | 

11. At the rate of $2.75 per day of 10 hours, how much 
should be given a man that works from a quarter before 8 until 
5 minutes past 11? 


12. If a railroad train travels 18 miles in 40 minutes, how 
far will it travel, at the same rate, in 74 hours? 


REVIEW. 


SPECIAL DRILLS. 


784. Give sums: 


163 + 137 
256 + 184 
149 + 312 
458+ 197 


42+-35-+ 77 
63 + 19+ 54 
87 + 22+ 48 
91+ 63+ 17 


785. Give differences: 


400 — 163 
501 — 875 
275 — 137 
650 — 488 


185 + 546 


668+ 193 ~ 


167 + 734 
476 + 155 


4,170 + 470 
1,260 + 850 
2,140 + 680 
3,450 + 390 


540 — 384 
361 — 149 
455 — 358 
662 — 176 


7,310 — 6,850 
8,610 — 7,680 
5,000 — 4,670 
4,960 — 4,380 


618 — 495 
455 — 128 
648 — 509 
856 — 147 


786. Give products: 


11x 15 
12x 14 
Se 
14x 14 


48 x 162 
32 X 374 
24 x 624 
36 X 662 


787. Give quotients: 


165 +15 
168 + 14 
169 +138 
294 + 14 


6162 + 162 
8374 + 3874 
7334 + 334 
6874 + 624 


788. Give answers: 


132 x5 
142 x4 
14,3, x 7 
137, x 8 


on 
= 
lon 


a 

= 

aloo O 
op a b& 


Coas 
oolbo crib 
| 


21 x 15 
22 x 14 
dl x 18 
41x 14 


185 + 45 
136 + 68 
220 + 44 
196 + 49 


24 + 22 
18 + 41 
49 + 1% 
35 + 34 


28 < 75 
40 x 874 
39 x 832 
49 x 25 


9334 + 662 
975 +75 
6123 + 874 
925 +25 


21 x 3% 
22 x.4,% 
18 x 32 
17 x 44 


388 ARITHMETIC, 


789. Oral Problems, 
1. How many ounces in 11,3, lb.? 
2. 258 yd. equal how many ft.? 


3. A dealer bought 652 tons of coal and sold 476 tons. 
How much had he left? 


4. Sold my wheat for $347 and my oats for $154. How 
much did I receive for both? 


5. 402 yd. of ribbon are cut into 7 pieces. Find the length 
of each piece. 


6. How many sq. yd. in a-floor 52 yd. long and 54 yd. wide? 
7. What will be the cost of 14 Ib. of lard at 14¥ per lb.? 
8. At 11 each, how many lead pencils can I buy for 27¢? 


9. What part of a 196-lb. barrel of flour is contained in a 
49-lb. bag? 


10. At 45¥ per yd., how much lace can be bought for $1.35? 


11. A woman has saved $8383. How much more must she 
save to have $1,000? 


12. What will be the cost of 16 Ib. of sugar at 43¢ per Ib.? 


13. Spent $2.56 for dry goods and $1.84 for groceries. How 
much did I spend for both? 


14. Find the cost of 8 lb. 10 oz. of butter at 32¢ per lb. 

15. At $.875 per yd., how much ribbon can be bought for 
p.75? 

16. Ifit takes 1% yd. of cloth to make a jacket, how many can 
be made from a piece of cloth containing 30 yd. ? 


17. A boy paid 50¥ for the use of a boat for 34 hours. What 
was the price per hour? 


18. If 13 pounds of raisins cost $1.69, what is the cost of 1 
pound? 


REVIEW. 


TABLE. 


339 


FOREIGN COMMERCE OF THE UNITED STATES. 


Exports AND Imports, 1875-1891. 


790. The following table shows the values of the exports and 
the imports of merchandise during each year from 1875 to 1891, 


inclusive. 


IMPORTS AND Exports oF MERCHANDISE. 


Year ending 


June 30. 


Exports. 


Imports. 


Excess of 
exports. 


Excess of 
imports. 


_— | |s S —  —_ | e 


1875 
1876 
1877 
1878 
1879 
1880 
1881 
1882 
1883 
1884 
1885 
1886 
1887 
1888 
1889 
1890 
1891 


Total. 


$513,442,711 
540,384,671 
602,475,230 
694,865,766 
710,439,441 
835,638,658 
902,377,346 
750,542 257 
823 839,402 
740,513,609 
742. 189,755 
679,524,830 


716,183,211 © 


695,954,507 
742,401,375 
857,828,684 


884,480,810 


$533,005,436 
460,741,190 
451,323,126 
437,051,532 
445,777,775 
667,954,746 
642,664,628 
724,639,574 
723,180,914 
667,697,693 
577,527,329 
635,436,136 
692,319,768 
723,957,114 
745,131,652 
789,310,409 


844,916,196 


Find the excess of exports or of imports for each year. 
Find the total exports and the total imports for the 17 years 
ending June 30, 1891, and the difference between the excess of 


exports and excess of imports for the same period. 


340 ARITHMETIC. 


SHORT METHODS. 


791. 4,846 7,854 
x 3% 274 
14,588 8 times 54,978 7 times 
2,9072 + of 3 times 6,108 4 of 7 times 
Multiply by 3, then add ¢ of this 218,1665 Ans. 
product. 
1. 247 x 44 6. 6,305 x 95% 
2. 1,896 x 53 7. 8,762 x 233 
3. 1,234 x 654 8. 12,3845 x 322 
4. 3,742 x 745 9. 7,890 x 1053 
5. 4,053 x 74 10. 67,890 x 2344 


792. Multiply 8,654 by 99. 
865,400 


Subtract the number from 100 times 


BEGITABI UNS: tis mia 

11. 7,885 x 99 21. 2,684 x 25 

12. 9,427 x 99 22. 9,321 x 334 
13. 6,073 x 99 23. 8,693 x 124 
14. 5,483 x 99 24. 4,862 x 662 
15. 2,761 x 999 25. 3,025 x 374 
16. 8,305 x 999 26. 3,464 x 622 
Lie ogo x 1 oe 27. 4,872 x 874 
18. 999 x 3,859 28. 860 x 24 

19. 9,832 x 990 29. 6,318 x 162 


20. 7,543 x 990 30. 9,204 x 75 


LONG TON. 341 
793. Divide. Do not write products (Arts. 385, 616): 
31. 41,874,365 + 9,999 33. 300,200,100 + 31,416 
32. 123,456,789 + 1,987 34. 254,637,809 = 26,543 
35. 837,029,456 + 16,074. 


794. Write answers: (Art. 385.) 


e 23146 38. 11223 40: 98643 49. 633386 
7809 1984 28643 16895 
68400 72063 87631 72084 

37. ——— 39. 41... 43. ——— 
o709 5999 17025 10203 


795. Avoirdupois Weight. Long Ton. 


In selling iron, coal at the mines, ores, etc., and in calculating the duties 
at the U. 8. custom houses upon imported goods, the following table is used: 


28 pounds (lb.) 1 quarter (qr.) 
4 quarters 1 hundredweight (cwt.) 
20 hundredweight 1 ton (T.) 


lewt.=112 1b. 1 T. = 2,240 lb. 
796. The ton of 2,240 pounds is called a long ton. Unless otherwise 


specified in a problem, the cwt. of 100 lb. and the ton of 2,000 lb. are to be 
taken. 


1. Reduce 25 'T. 13 cwt. 2 qr. 25 lb. to pounds. (Long ton.) 

2. Change 100,000 pounds to tons (long), cwt., qr., lb. 

3. Find the duty at 1)5% per lb. on an invoice of tin weigh- 
ing 33 T. 7 ewt. 20 lb. (Long ton.) 

4. What is the total weight in tons (long), etc., of 19 barrels 
of soda-ash weighing 13 cwt. 2 qr. 10 lb. each? 

5. Find the weight of the rails required for 100 miles of 
double track (four rails), the weight of a rail being 18 pounds 
per running foot. 

What will be the cost of the rails at $21 per long ton ? 


6. A coal dealer buys 175 (long) tons of coal. How much 
does he receive for it at $5 per ton of 2000 pounds? 


342 ARITHMETIC. 


MEASUREMENTS. 


800. What is the length in inches of a row of four envelopes, 
each five inches long, placed end to end? What is the length in 
feet and inches? 


1. What is the width in inches of four such rows just touch- 
ing each other? What is the width in feet? 
How many envelopes are there? How many sq. in. in each 
envelope? How many sq. in. are covered by all of them? 
2. How many envelopes 5 inches by 3 inches would cover 
the top of a table 4 feet 2 inches long and 2 feet 6 inches wide? 


3. Draw a rectangle to represent a floor 24 feet long 18 feet 
wide. Draw rugs 6 feet long, 3 feet wide, and see how many 
will be needed to cover the floor. 


4. How many boards 12 feet long, 6 inches wide will be 
required for a floor 8 yards long, 6 yards wide? 

If the boards run lengthwise, now many boards in length are used? How 
many boards wide? 

5. Given the area in square inches of a surface to be covered 
with envelopes, and the area in square inches of an envelope, 
how is the number of envelopes ascertained ? 

If the area of the surface to be covered is given in square feet, 
how must we proceed ? 


MEASUREMENTS. 343 


6. Given the dimensions of a surface to be covered and the 
dimensions of the articles to be used for covering, how can we 
indicate the operations to be performed, without actually doing 
the work? 


How about the denominations used ; yards, feet, etc. ? 


7. How many bricks 8 inches by 4 inches will be needed for 
a walk 24 yards long, 6 feet wide, making no allowance for 
waste ? 


First indicate the work. Cancel. 


8. How many paving tiles } foot square will cover a hearth 
6 feet long, 3 feet wide? 


Make a diagram. 


9. How many boards 12 feet long, 9 inches wide will be 
required for a close fence 120 yards long, 6 feet high? 


10. Find the number of paving stones 9 inches by 3 inches, in 
a street 100 rods long, 10 yards wide. 


11. A man buys a piece of ground 300 feet long, 150 feet 
wide. He builds a house, 50 feet by 30 feet, and a shed 12 feet 
by 13 feet. How many square yards will he have left for a 
garden ? 


12. There are 160 square rods in an acre. How many square 
yards are there in an acre? 


13. Give the dimensions, in yards, of a field that will contain 
just an acre. Of one that will contain two acres. 


14. Draw a rectangle 2 inches x 3 inches. Draw one twice 
the size. What are the dimensions of the latter? 

A plot 100 ft. x 100 ft. is how many times as large as a plot 
20 ft. x 25 ft.? 


15. How many square feet are there in a fence 10 feet high 
around a lot 250 feet long, 200 feet wide? 


344 ARITHMETIC. 


16. The owner of a piece of ground 250 feet long, 200 feet 
wide, takes 10 feet from each side to make a gravel walk, and 
uses the remainder for a garden. Give the dimensions of the 
garden and its area in square feet? How many square feet are 
taken up by the walk? How many square feet in the whole 
piece of ground ? 


17. How many square feet of flagging would be required for 
a sidewalk 10 feet wide outside a lot 250 feet long, 200 feet wide? 


If 250 running feet of sidewalk, 10 feet wide, were laid on two sides of 
the lot, and 200 running feet on each of the other sides, would the job be 
finished ? 


18. At $80 per acre what is the value of a field 80 rods long, 
70 rods wide? 
What will it cost to fence the field at 20% per running yard? 


19. A room is 24 feet long, 18 feet wide, 12 feet high. Draw, 
touching each other, four rectangles representing the four walls. 
Write the dimensions of each wall. 


What are the dimensions of the large rectangle made up of the four 
smaller ones? Give the area in square feet. In square yards, 


e 


TIME BETWEEN DATES. 345 


20. Show by a diagram the shape of a piece of paper that 
when folded will entirely cover a box 12 inches long, 6 inches 
wide, 4 inches high. Write the dimensions. 

801. This is called the “development” of the box. 

What is the area of the paper in square inches? 

21. Make a diagram of a room 24 feet long, 18 feet wide, 12 
feet high, showing the surface that is generally plastered. 

How many square yards of plaster will be needed for the 
above room, making no allowance for doors, windows, etc. ? 

22. What is the length of a rectangular field 60 rods long 
that contains 60 acres? 

23. To contain 48 square yards, how many yards long must 
be a piece of carpet 27 inches wide? 

24. I have bought 24 yards of dress goods, 27 inches wide. 
How many square yards does the piece contain? 

How many yards of lining 32 inches wide will contain the 
same number of square yards? 


24 yards long. ? yards long. 


# yd. | 18 sq. yd. | = | 18 sq. yd. § yd. 


25. How many square yards are there in 27 rugs, each 63 
inches long, 45 inches wide? 


TIME BETWEEN DATES. 
802. Oral Problems. 
1. How many hours from 8 o'clock Saturday afternoon to 
9 o'clock Sunday morning? 
2. How many days from May 1 to June 1? 
3. A boy takes a spoonful of medicine every hour. If he 


takes the first dose at 2 o’clock, at what hour will he take the 
sixth? The second? The fourth? 


4. A man begins work on the morning of the 6th and ends on 
the evening of the 11th. How much does he earn at $3 per day? 


346 ARITHMETIC. 


6. An importer receives some cases of goods numbered con- 
secutively. How many cases are there if the lowest number is 
29 and the highest number is 53 ? 


6. How many posts 6 ft. apart will be needed for a fence 
120 ft. long? For a fence 6 ft. long? 12 ft. long? 


7. Find the time from Jan. 1 to Jan. 31, counting the first 
and the last day. Omitting both days. 
8. How many days from July 4 to August 15, inclusive ? 
9. How many chapters from the 25th to the 49th, exclusive? 
10. A girl begins at the 146th problem and solves all those 


on two pages. If the last is the 172d problem, how many does 
she solve ? 


803. In finding the time between dates, either the first or the 
last day is excluded; that is, from the lst to the 21st is con- 
sidered 20 days. 

How many days from March 4 to Sept. 1? 


March 4 to March 31, 27 days. 


Excluding March 4, there remain in se ea os m 
the month 31 —4, or 27 days. To this add ay 
the number of days in April, May, June, June Suis 
July, and August. Since March 4 is ex- July Sea 
cluded, we take 1 day in September, mak- Aug. SLU 
ing the total 181 days. Sept. Tene: 


Ans. 181 days. 
804. How many days from 7 


1134 Jan: | to Febalor 16. Feb. 29 to April 1? 
12. Oct. 31 to Dec. 30? 17. May 21 to July 4? 
13. Sept. 30 to Dec. 16? 18. April 7 to May 27? 
14. Noy. 1 to Dec. 19? 19. June 10 to Aug. 1? 


15. March 16 to April 25? 20. July 4 to Sept. 1? 


TIME BETWEEN DATES. 347 


805. Slate Problems, 

(Take note of leap year.) 

How many days from: 

Feb. 6, 1892, to Oct. 1, 1892? 
Oct. 14, 1892, to March 8, 1893? 
Jan. 1, 1892, to April 19, 1892? 
Dec. 23, 1891, to March 8, 1892? 
Sept. 3, 1892, to Feb. 1, 1893? 
March 16, 1892, to Dec. 25, 1892? 
June 8, 1892, to Nov. 29, 1892? 
Aug. 17, 1892, to Jan. 3, 1893? 
April 4, 1892, to July 4, 1892? 
May 16, 1892, to Oct. 14, 1892? 


_ 
fe 


11. How much wages at $4 per day should a man receive 
rom Monday, Jan. 2, 1893, to Feb. 28, inclusive, no pay to be 
eeceived for Sundays or Washington’s birthday? 


12. A man borrowed $100 April 4, and returned it November 
25. How many days’ interest did he owe? (Do not include 
both days.) 

13. May 1, 1898, fell on Monday. Upon what day of the 
week did July 4 fall? | 

14. How many days does vacation last if it begins on the 
morning of Saturday, July 2, and school commences on the first 
Tuesday of September ? 

15. A man borrows some money June 16, and agrees to return 
it in 60 days. On what date should he pay it? 

16. A traveler starts upon a trip Aug. 24, 1894, and reaches 
home again Feb. 10, 1895. How long is he away? 


348 ARITHMETIC. 


806. When the difference between dates is more than a year, it is cus- 
tomary to take 30 days to each month. (See Appendix, Part IIT.) 


Find the difference in time between March 8, 1879, and Jan. 1, 
1898. | 


1893 1 1 
Writing 1893, Ist month, Ist day, we subtract jong 9 98 
8 


1879, 3d month, 3d day. "1S a ee 
Answer, 13 years 9 months 28 days. 


17. George Washington was born Feb. 22, 1732. How old 
was he at the signing of the Declaration of Independence, July 
4, 1776? 

18. Abraham Lincoln was first inaugurated president, March 
4, 1861. How long had he served at his death, April, 15, 1865? 

19. The battle of Lexington took place April 19,1775. The 
treaty of peace was signed Sept. 3, 1783. How many years, 
months, and days between the two events ? 

20. How many years elapsed between the discovery of America 
by Columbus, Oct. 12, 1492, and the landing of the Pilgrims, 
Dec. 21, 1620? 

21. General Harrison fought the Battle of Tippecanoe Nov. 
7, 1811. He was inaugurated president 29 years 3 months 27 
days later. Give the date of his inauguration. 

22. How long was it after the treaty with England, signed 
Dec. 24, 1814, that the Mexican treaty was concluded, Feb. 2, 
1848? 

23. General Taylor died July 9, 1850. How long did he live 
after the capture of Monterey, Sept. 24, 1846? 

24. President Garfield was born Noy. 19, 1831. How old 
was he at his inauguration, March 4, 1881? 

25. The last battle of the Mexican war took place Sept. 14, 
1847. The Battle of Bull Run was fought 13 years 10 months 
7 days later. What was the date of this battle? 

26. Find the time between July 4, 1776, and Jan. 1, 1894. 


INTEREST. O49 
PERCENTAGE. 
807. Oral Exercises. 
1. Find 4% of $125. 6. 334% of 1 day. 
2. 25% of 16. 7. 622% of $12. 
3. 6% of 200 cows. 8. 9% of $23. 
4. 1% of 150 lb. 9. 75% of 3 gal. 
5. 20% of 65 yd. 10. 14% of $400. 


808. Slate Exercises. 


1. Find 6% of $576. 9. 25% of $156. 

2. 41% of $340. 10. 1% of $156. 

3. 25% of 1,876 bu. 11. 4% of $156. 

4. 121% of 864 cows. 12. 50% of 480 hr. 

5. 50% of 482 yd. 13. £% of 480 hr. 

6. 331% of 576 soldiers. 14. 1% of $1,420. 

7. 162% of 696 gal. 15. 31% of $66. 

8. 61% of $4.96. 16. 74% of 360 days. 
INTEREST. 


809. In computing interest, the year is considered as com- 


posed of 12 months of 80 days each. 


810. Oral Exercises. 
Find the interest on: 
1. $90 for 2 mo. at 4%. 


2. $60 for 60 da. at 6%. 


3. $100 for 2yr.6 mo. at 5%. 


4. $120 for 30 da. at 5%. 
5. $300 for 90 da. at 3%. 


6. $100 for 1 yr. 3 mo. at 4%. 


7 


. $50 for 3 yr. at 6%. 


8. $100 for 2 yr. 4 mo. at 6%. 


9 
10 


. $60 for 40 da. at 6%. 
- $120 for 120 da. at 5%. 


350 ARITHMETIC. 


811. Find the interest on $63 for 4 yr. 5 mo. at 5%. 
p69 is called the principal. 
5 = rate. 
4 yr. 5 mo. = time. 


B12 i Tne vse principal X rate < time (in years) 


100 
4 years 5 months = 4,5 yr. = 23 yr. 
21 
BOB x 2 x 28 $95.60 _ gig 9) 4 Ans. 


AO We 4 
4 


Notre. — The divisor 100 is canceled by placing a decimal point before 21. 
Find the interest on $160.50 for 3 mo. 15 da. at 6%. 


OU29 6 7 56175 


160.50 x —~ ne : = 1.89050 x 95 = 2.808 + 


0p ya 2 Ans. $2.81. 
2 
Find the interest on $69.75 for 1 mo. 17 da. at 4%. 


00775 4 AT 
0669075 x —— = .86425. Ans. 36 cents. 


100 * 360 
20 
813. Slate Exercises. 
Find the interest on: 
1. $192 for 3 yr. 7 mo. at 5%. 
2. $60 for 2 mo. 12 da. at 4%. 
3. $240 for 1 yr. 1 mo. at 6%. 
4. $14.40 for 5 yr. 5 mo. at 5%. 
5. $36 for 77 days at 44%. 


REVIEW. BOL 


$99 for 21 months at 6%. 

$192 for 2 yr. 4 mo. at 5%. 

$600 from Jan. 1 to Jan. 16 at 4%. 

$1,200 from July 1, 1891, to Jan. 1, 1893, at 6%. 
10. $57.60 from Oct. 4, 1890, to Feb. 4, 1894, at 5%. 


eo Oo ~~ OH 


REVIEW. 


814. Oral Problems. 
1. What part of 4is $? (16 twentieths, 15 twentieths.) 
2. 16 is how many hundredths of 64? 
3. What per cent of 25 is 5? 
4. What part of 2 lb. 1 oz.is11b.? (83 oz., 16 oz.) 
5. Divide 4 gal. by 3 pt. 


6. How many pencils at 4 mills each can be bought for a 
dollar ? 


7. Write 5 as a decimal. 
8. Divide 34 by 200. 


9. At 20¢ per qt., what will be the cost of 2 gal. 8 qt. 1 pt. 
of maple syrup ? | 


10. Find the cost of 4 T. 400 lb. of coal at $5 per ton. 


11. A man puts 4 |b. 8 oz. of tea into 12 oz. packages. How 
many packages does he make ? 


12. 4 pecks 3 quarts of apples are given to some children. If 
each child’s share is 5 quarts, how many children are there? 


13. If it takes 3 hours 20 minutes to hoe a row of corn, how 
many rows can a man do in 2 days of 10 hours each? 


352 ARITHMETIC. 


14. How many square inches in the surface of a sheet of paper 
1 ft. 8 in. long, 11 in. wide? 


15. How many pieces of paper 2 inches square will exactly 
cover a slate 12 inches long, 8 inches wide? 


815. Slate Problems. 
1. What part of 6 hr. 17 min. 5 sec. is 38 hr. 15 min. 25 sec. ? 


2. Ifaman walks at the rate of 3 mi. and 96 rd. per hour, 
how far will he walk in 3 hr. and 20 min. ? 


3. What is one-ninth of 28 bu. 3 pk. and 7 qt.? 


4. Three men buy a house for $1,200. A furnishes $600; 
B, $400; C,$200. They sell the house for $1,500. How much 
money should each receive? 


5. If 5 T. and 1,000 Ib. of coal cost $30.25, how much will 
be paid for 7 T. and 320 lb. ? 


6. At 25¢ per hour, how much should a man receive that 
works 8 hours and 86 minutes? 


7. At $45 per month, what is the rent of a house for 2 yr. 
7 mo. and 8 da. ? 


8. If 2 1b. 4 0z. of tea cost $1.35, what will be the cost of 
1) Ib: 12 oz.? 


9. How many sq. in. in a paving tile 6 in. square? How 
many sq. in. in a rectangle 4 ft. by 3 ft.? How many paving 
tiles 6 in. by 6 in. would cover a surface 4 ft. by 3 ft. ? 


10. A merchant imports 360 yd. of dress goods, 27 inches 
wide, costing 30% per yd. What will the duty be at 8% per 
square yard, and 50 per cent of the cost in addition ? 


11. A man pays $60 interest per year. How much interest 
does he pay in 3 yr. 7 mo. 10 da.? 


12. Find four-ninths of 28 bu. 3 pk. 7 qt. 


MEASUREMENTS. 353 


APPROXIMATIONS. 


816. Give an estimate of the answer (Art. 521): 


1. If3 T. and 1,988 lb. of coal cost $19.97, what will be the 
cost of 8 T. and 1 1b.? (Nearly 4 tons cost nearly $20.) 


2. At $500 per year, what will be the rent of a house for 
1 year 11 mo. and 29 da.? (Nearly 2 years.) 


3. Find the cost of 5 bbl. sugar, averaging 299 lb. each, at 
415 ¢ ner lb, 


4. What is the interest on $199.86 at 6%, for 5 mo. 28 da. ? 


5. If 11 men and 2 boys can finish a piece of work in 234 
days, how long would it take 23 men and 5 boys? 


6. What decimal of 639 acres is 321 acres? 
7. What will be the cost of 20,060 bricks at $19.90 per M.? 


8. A farmer sells 5,584 lb. of rye at 87% per bu. of 56 |b. 
How much does he receive? 


9. If 9 1b. and 15 oz. of tea cost $7.95, what will be the cost 
of 21 lb. and 1 oz.? 


10. Paid freight on 1,987 lb. at 70% per hundredweight. 
How much did I pay ? 
MEASUREMENTS. 
817. Make diagrams when necessary. 


1. A man has a lot 100 feet by 200 feet. How many square 
feet will he have left for a garden after he builds a house 25 feet 
by 60 feet? 


2. One wall of a room is 24 feet long and 12 feet high. 
There is a door in it 8 feet high, 44 feet wide. How many 
square yards of plastering will be needed to cover the wall? 


354 ARITHMETIC, 


3. A brick is 8 inches long, 4 inches wide, 2 inches thick. — 
How many square inches are there in the surface of the widest 
face? In the surface of one side? In the surface of one end? 


4. How many bricks laid on the widest face will be needed 
for a walk 288 inches long, 96 inches wide? 


5. How many bricks laid on the side will be needed for a 
walk 24 feet long, 8 feet wide? 


6. Make a diagram of a piece of paper that when folded 
will just cover the six faces of a brick 8x42 inches. How 
many square inches of paper would be needed ? 


7. How many square feet are there in a roll of wall paper 
24 feet long, 18 inches wide? 


8. How many rolls 24 feet long, 14 feet wide, would be 
required to paper the ceiling of a room 45 feet long, 36 feet 
wide, making no allowance for matching or waste? 


9. The owner of a piece of ground 200 feet wide, 300 feet 
long, divides it into lots 25 feet by 100 feet. How many lots 
are there ? 


10. Make table of: 


818. Square Measure. 
square inches (sq. in.) 1 square foot (sq. ft.) 


square feet 1 square yard (sq. yd.) 

square yards 1 square rod (sq. rd.) 
160 square rods 1 acre (A.) 

acres 1 square mile (sq. mi.) 


11. The owner of a piece of ground 600 feet long, 150 feet 
wide, builds a fence 6 feet high around the plot. How many 
square feet of fence are there? 


12. What would be the cost of building 1,800 feet of fence 6 
feet high at $1 per square yard? 


MEASUREMENTS. 350 
13. A farm is one mile square. How many 40-acre fields 
does it contain? 


14. How many yards of fence will be needed to enclose the 
plot of ground shown in the following diagram ? 


12 rods 


15. The above field was originally a rectangle, but the owner 
sold one piece 5 rods by 3 rods, and a second piece 3 rods by 7 
rods. How many square rods did it contain at first? What is 
its present area? 


16. How many acres in a field in the shape of a triangle 
whose base and perpendicular measure 40 rods each? 


17. Calculate the number of 
square yards in the field shown 
in the accompanying diagram. 


18. The owner of a field 160 


yards long, 121 yards wide sold |. 24 yd. (6 ya. 
from one corner a triangular piece 

40 yards long, 801 yards wide. Find the number of square 
yards in the part remaining. (Make diagram.) 


15 yd. 


I 
s 
l 
{ 
J 
! 


356 ARITHMETIC. 


19. How many acres are there in a triangular plot of ground 
when the base measures 80 yards and the perpendicular measures 
604 yards? 


20. What are the dimensions of the box that can be made of 
a piece of paper of the size shown in the following diagram ? 
How many square inches of paper are needed for such a box 
(making no allowance for pasting) ? 


4 in. 


8 in. 


DENOMINATE NUMBERS. (Long Measure.) 


819. Slate Exercises. 
1. Change 43 yd. to rods and a fraction. 
2. Change 43 yd. to rods and yards. 
3. Change 43 yd. to rods, yards, and feet. 
4. Change 43 yd. to rods, yards, feet, and inches. 
5. Change 72 yd. to rods, etc. 
6. Change 66 yd. to rods. 


DENOMINATE NUMBERS. 357 


820. Change to rods, etc.: 


7. 49 yd. 14. 1,836 in. 
8. 147 ft. 15 ean 
9. 1,764 in. 16. 52 yd. 
10. 812 rd. 17. 492 yd. 
TLS o0-yd: 18. 49 yd. 1 ft. 6 in. 
12. 51 yd. 19.. 1484 ft. 
13. 153 ft. 20. 1,782 in. 


821. Change 1,581 in. to rods, ete. 


12) 1581 in. 
3)131 ft. 9 in. 
4)43 yd. 2 ft. 9 in. 
2a 
ine 
7rd. 9 half-yards 2 ft. 9 in. 
7 rd. 43 yd. 2 ft. 9 in. 


Grd. eva *2, thai din, 
+ 1ft. Gin. =#4 yd. 


7rd. Syd. 1ft. 3in. Ans. 


822. To find how many rods in 43 yards, we divide by 53. 53 yards = 
11 half-yards. 43 yards = 86 half-yards. As there are 11 half-yards in a 
rod, 86 half-yards will be equal to 7 rods and 9 half-yards, or 7 rods 44 
yards. Changing } yd. to 1 ft. 6 inches, we obtain the answer as above. 


823. Change to rods, etc.: 


21. 1,488 in. 24. 2,796 in. 27. 3,453 in. 
22. 984 in. 2a Lokal try: 28. 1,278 in. 
28. 1,345 in. 26. 1,470 in. Boul oO hon: 


30. Change 5 rods to inches. 

31. Change 990 inches to rods. 

32. How many inches in 7 rods 1 yd.? 
33. Change 1,422 inches to rods. 


358 ARITHMETIC. 


824. Add: 

34. 4rd.3 yd. 1 ft. 36. 5rd. 4 yd. 2 ft. 
9rd. 4 yd. 2 ft. 5 yd. 1 ft. 
3 rd. eet Ger Gird lavas 

BERANE Recah S7 hl rd26 yonder 
3 rd. lsat: 4 rd. Aly § 

2 yd. 2 ft. 5 yd. 1 ft. 

4 rd. ger 6 rd. 


38. From 8rd. 1 ft. take 2 rd. 2 ft. 

39. Find the difference between 3 rd. 1 yd. 1 ft. and 16 rd. 
40. Multiply 5rd. 4 yd. 2 ft. by 4. 

41. Multiply 11 rd. 2 ft. by 10. 

42. Divide 30 rd. 5 yd.2 ft. by 8. 

43. Divide 34 rd. 2 yd. by 9. 


SOLID CONTENTS. 


825. How many one-inch cubes can be placed on the bottom 
of a box 3 in. long, 4 in. wide? 


1. If the box is one inch high, how many will it hold? If 
the box is 2in. high? 3 in. high? 


2. A cube one inch long, one inch wide, one inch high con- 
tains a cubic inch. 

How many cubic inches in a box 38 in. long, 4 in. wide, 1 in. 
high? Ina box 8 in. long, 4 in. wide, 2 in. high? In a box 4 
in. long, 4 in. wide, 4 in. high? 

3. If you had 24 one-inch cubes, how could you pile them 
to make a solid with six rectangular faces ? 

If the pile was 2 inches high, how many cubes would there 


be in each tier? How many square inches would the lower tier 
cover? 


SOLID CONTENTS. 359 


How could the 24 cubes be arranged to make a pile 3 inches 
high? 
4. Can you give a rule for finding the contents of a box 6 in. 
long, 8 in. high, 4 in. wide? 


5. How many cubic inches of water would a tin box hold, 
the dimensions of the box being 5 in. by 34 in. by 4in.? 


6. How many one-inch cubes could be placed in a box one 
foot long, one foot wide, one foot high ? 


7. How many cubic inches in a cubic foot? 


8. How many one-foot cubes could be placed in a cubical 
box one yard long, one yard wide, one yard high? 
How many cubic feet in a cubic yard? How many cubic 
inches in a cubic yard? 
9. How many cubic inches in a solid, 3 yd. long, 2 ft. wide, 
6 in. high? How many cu. ft.? How many cu. yd.? 
Indicate the operations necessary in each case to obtain the 
correct answer. 
What should be done with the denominations in each case 
before beginning the work of obtaining the solid contents? 


10. A cord of wood contains 128 cu. ft. If the wood is cut 
into 4-ft. lengths, what should be the other two dimensions of a 
regular pile to hold just a cord? 


11. How many cubic feet of air in a room 24 ft. long, 18 ft. 
wide, 12 ft. high? 

12. A gallon contains 231 cu.in. Give the dimensions of a 
tin box that will hold exactly a gallon. 


13. Find the solid contents of a piece of timber 25 ft. long, 
8 ft. wide, 5 ft. thick. Is it larger or smaller than a piece 4 ft. 
wide, 4 ft. thick, and 28 ft. 6 in. long? 

14. How many cubic yards of earth will have to be removed 
in digging a cellar 18 ft. wide, 55 ft. long, 6 ft. deep? What 
will be the cost at 60% a load (1 cu. yd.)? 


360 ARITHMETIC. 


15. Give the width of a wagon body 18 in. high, 6 ft. long, 
that will hold, when full, a cubic yard. 


16. About how many gallons are there in a cu. ft.? 


17. A bushel contains 2,150.4 cu.in. About how many cu 
ft. in a bushel? 


18. Find (by cancellation) the capacity in gallons of a tank 
| ft. 9 in. long, 1 ft. 3 in. wide, | ft. 10 in. deep. 


19. Find (by cancellation) the capacity in bushels of a bin 1 
yd. long, 2 ft. 4 in. wide, 5 ft. 4 in. high. 


20. How many cords of wood (128 cu. ft.) are there in a pile 
24 ft. long, 4 ft. wide, 12 ft. high? 


21. A brick is 8 in. long, 4 in. wide, 2 in. thick. How many 
bricks are there in a pile 90 bricks long, 60 bricks wide, 30 
bricks high? What are the dimensions of the pile? 


How many cu. in. in 1 brick? In the pile? 


22. Find the number of bricks in a wall 24 ft. wide, 48 ft. 
high, 1 ft. thick, making no allowance for mortar, ete. 


23. How many bricks are there to a cu. ft. ? 


24. Allowing 20 bricks to a cubic foot when laid in mortar, 
how many bricks will be needed for a wall 24 ft. wide, 50 ft. 
high, 20 in. thick ? 


25. What will be the cost of building a stone wall 40 rods 
long, 4 ft. high, 1 yd. thick, at $6.40 per perch of 243 cu. ft.? 


APPROXIMATIONS. 


826. Give sight answers in-whole numbers: 


1. If there are about 74 gal. to a cu. ft., estimate the number 
of gallons in a tank 5 ft. long, 3 ft. wide, 4 ft. high. 


2. If there are about 14 cu. ft. in a bushel, estimate the con- 
tents in bushels of a bin 5 ft. x 3 ft. x 4 ft. 


DENOMINATE NUMBERS. 361 


3. Give the dimensions of a tank of 150 gal. capacity. 
4. Give the dimensions of a bin that will hold 100 bushels. 


5. At 20 bricks laid in mortar to the cu. ft., give the length 
and the height of a wall 1 ft. thick that can be built with a 
thousand bricks. 


6. At $1 a load (1 cu. yd.), give the dimensions of an 
excavation that can be made for $100. 


7. A cu. ft. water (about 74 gal.) weighs 624 lb. About 
what does a gallon weigh? <A pint? 


8. If iron is about 71 times as heavy as water, about what 
does a cubic foot of iron weigh? 


9. About what is 492% of $801? 


10. About what will be the interest at 6 per cent on $100 
for 3 yr. 11 mo. 29 da.? 


827. Cubic Measure. 


ty Boos Clg Als ak Cte ite 
Ze Pech itel hk: Cl ya 
231 cu.in. 1 gallon. 
2,150.4 cu.in. 1 bushel. 

128; cu.ft. 1 cord: 


828. Troy Weight. 


In weighing gold and silver, troy weight is used. The following is the 
table: 
24 grains (gr.) 1 pennyweight (dwt.) 
20 pennyweights 1 ounce (0z.) 
12 ounces 1 pound (1b.) 


The abbreviation for pennyweight is also written pwt. 


829. The pound apothecaries’ weight —used in putting up prescrip- 
tions — contains the same number of grains (5,760) as the pound troy. 


830. The pound avoirdupois contains 7,000 grains, 


362 ARITHMETIC. 


1. At $18 per M., what will be the cost of a pile of bricks 
20 ft. long, 15 ft. wide, 6 ft. high? <A brick measures 8 in. by 
4 in. by 2 in. 

2. Find the cost of a mlves pitcher, weighing 5 lb. 5 oz. 12 
pwt. 12 gr., at 48 cents per ounce troy. 

3. How many pounds and ounces avoirdupois does the above 
pitcher weigh ? 

4. How many 5-grain pills can be made from an avoirdupois 
pound of quinine? From a pound apothecaries’ weight? 

5. How many spoons weighing 18 pwt. 16 gr. each, can be 
made from 11 oz. 4 pwt. of silver ? 


CANCELLATION. 
834. Indicate operations, and cancel where possible. Terms compared 

should be of the same denomination. 

1. If 90 tons of coal cost $472.50, what will be the cost of 
132 tons? 

2. If3 lb. and 4 oz. of tea cost $1.95, what will 12 oz. cost? 

3. A party of men can build 16 rd. 2 ft. of wall in 20 days. 
How long will it take them to build 4 yd. and 6 in.? 

4, What will be the cost of 8 bu. 2 pk. 7 qt. 1 pt. of oats if 7 
bu. 1 qt. cost $4.50? 

5. By traveling at the rate of 20 miles a day, a person can 


complete a journey in 18 days. At what rate must he travel to 
finish it in 15 days? 


REVIEW. 
841. Add across: 
1. 135 -+ 163+ 8 6. 59% + 35 + 43 
2) 441.594 978 7. 78+182+ 404 
38. 193 + 33+ 354 8. 8524+5114 87, 
CMe, vile gon ae 4 9. 834914 25, 
5. 


238 4 544 32,5, 10. 6644+ 82+ 144 


842. Subtract across: 
1G 
12. 
13; 
14. 


15. 


843. Multiply. 


21. 
22. 
23. 
24. 


25. 


254 — 18,1, 
632 — 492 
70,4, — 154 
925 — 247 
831 — 157%, 


872 x BL 
488 x 44 
644 x 104 
292 x 64 


18% x 54 


844. Divide: 


31. 


32. 


33. 


34. 


35. 


13)2054 
14)1864 
15) 2504 
16)1984 


21)4504 


REVIEW. 


128 x 54 
634 

44 
63 


16. 
17. 
18. 
19. 


20. 


i 


t 


26. 
27. 
28. 
29. 


30. 


36. 


37. 


38. 


39. 


40. 


Cc Oo 


of aloo Ble 
| 
ie 
(Su) 
wt wh oO 


| 
oo 
~J 
cs 


| 
a 
on) 


Do not reduce to improper fractions: 


454 x 21} 
502 x 104 
383 x 114 


31)9704 
24)5588 
25) 5684 


32)9654 


36)7222 


363 


364 ARITHMETIC. 


845. Miscellaneous. 


1. Find the value of (a) (6.125 + 8.75 — 9.1235) + 0125; 
(6) (1,708.4592 = .00024) x .008. 


2. Simplify eee at -- 1th, 


3. (8 of J) + (Gof $) — ($ of 2)=? 
kof Th yy of 43 
Sof 15 tx Il 


5. Add 8§ -+ $#+- 2+ 4+. 
6. Find the value of 728 — 3—3— 
7 14x ($+2)x#4 


cleo 
| 
ool 


8. Reduce 7% of 2 of 3 of a to a decimal. 
9. If .1875 of a vessel cost $ 273.121, what is the value of 
3x of it at the same rate? 


10. A person owning 55; of a factory sells 75 per cent of his 
share for $1,710. What is athe value of the whole factory ? 


11. Find 3 of 2 days 5 hours 45 minutes. 
12. What is the interest on $760 for 5 months at 34 r7o? 


13. If a piece of cloth is 20 yd. long and 2 yd. broad, how 
broad is another piece which is 12 yd. long and oolitatit as many 
square yards as ae first ? 

1114 14-1 
aa ie Ex + re as Ea 

15. A merchant insures property worth $20,000 for % of its 
value. How much does he pay, the rate being 11%? 


14. Simplify 


16. If 7 men can do a piece of work in 104 days, how long 
will it take 8 men and 5 boys to do the same work, each boy 
doing one-half as much as a man? 


MEASUREMENTS. 365 


ANGLES, TRIANGLES, QUADRILATERALS. 


846. The following may be drawn free hand, the compasses being 
reserved for the geometrical problems in Chapter XVI. 


1. Draw two lines meeting at a point. 


These lines make an angle. 
2. Draw two lines that will make four angles. 


3. Draw two lines so as to make two angles. 


Two such angles are called adjacent angles. 


4. Make two equal adjacent angles. 


Equal adjacent angles are called right angles. A line making a right 
angle with another line is said to be perpendicular to it. 


5. Draw two lines so as to make one right angle. 


Is the right angle made by two lines, each 10 ft. long, any larger than 
a right angle made by two lines, each 1 inch long? 


6. Whatis the smallest number of straight lines that will enclose a space? 
Draw a figure enclosed by the smallest possible number of 
straight lines. What isits name? Why? 


7. Make a triangle having one right angle. 


8. Can you draw a triangle having two right angles? Why? What 
name is given to lines that will not meet, no matter how far they are 
extended? 


9. An angle less than a right angle is called an acute angle. 
Draw a triangle containing an acute angle. 
10. Can you draw a triangle containing two acute angles? 
Three acute angles? 
11. An angle greater than a right angle is called an obtuse angle. 
Draw a triangle containing an obtuse angle. 


12. Can you draw a triangle containing three obtuse angles? Contain- 
ing two? 


366 ARITHMETIC. 


13. Draw a triangle with sides 2 inches, 3 inches, 4 inches, 
respectively. 


A triangle having no two sides equal is called a scalene triangle. 


14. Draw a triangle having two equal sides. 


This is called an tsosceles triangle. The unequal side is called the base. 


15. Draw an isosceles triangle with the base uppermost. 
With the base on the left. On the right. 


16. Draw a triangle having three equal sides (an equilateral 
triangle). 
17. Draw asquare. Draw a rectangle 4 in. by 3 in. 


How many right angles in each ? 


18. Draw a four-sided figure having its opposite sides parallel, 
but containing no right angle (rhomboid). 

What kinds of angles does it contain? How many of each? Write 
name in each angle. 

19. Draw a four-sided figure, having all its sides equal, but 
containing no right angle (rhombus). 


20. Draw a quadrilateral (four-sided figure) having only two 
parallel sides (trapezoid). 


21. Draw a quadrilateral having no parallel sides (trapezium). 


22. Draw a rhombus, each side 2 inches. <A square, each side © 
2 inches. 
What is the difference between them? Which is larger? 


23. A parallelogram is a quadrilateral that has its opposite 
sides parallel. 


Name the parallelograms that have four equal sides (equilateral). Those 
that have four equal angles (equiangular). 


24. The height of a parallelogram is called its altitude. Draw 
a rectangle, base 34 inches, altitude 24 inches. Draw a rhom- 
boid, base 34 inches, altitude 21 inches. Draw several rhom- 
boids of the above dimensions, all differing in shape. 


MEASUREMENTS. 367 


25. Cut out of paper a rectangle, base 3 inches, altitude 2 
inches. Cut out a rhomboid, base 8 inches, altitude 2 inches. 
Place one upon the other, and see how their areas compare. 

26. Can you calculate the number of square inches in a rhom- 
boid whose base is 3 inches and altitude 2 inches? 

27. Draw a rectangle, base 4 inches, altitude 3 inches. Divide 
by a diagonal into two triangles. Mark in each triangle its area. 

28. Draw a right-angled triangle, base 4 inches, perpendicular 
(altitude) 3 inches. Calculate its area. 

29. Draw a rectangle, base 4 inches, altitude 3 inches. From 
the middle point of the upper base draw lines to the extremities 
of the lower base, making three triangles. Mark in each triangle 
its area. 

30. Draw an isosceles triangle, base 4 inches, altitude 3 inches, 
and calculate its area. 


847. Areas of Triangles and Quadrilaterals. 
Find the areas of the following: 


1. A right-angled triangle whose sides measure WY Ig 
15, 20, and 25 inches, respectively. 

2. A right-angled triangle whose base measures 64 yd., 
perpendicular 48 yd. 


3. A triangle whose base measures 18 rods, altitude 13 rods. 
4. A square whose side measures 36 feet. 97.{t. 
5. A rectangle 42 yd. by 387 yd. 


6. A rhombus whose base is 97 feet, 
altitude 63 feet. 1 TCO TL 


7. A rhomboid, base 33 meters, altitude 28 meters. 

8. A trapezoid whose parallel 
sides measure 1] and 16 ft., re- 
spectively, the perpendicular dis- 
tance between them being 6 ft. 


368 ARITHMETIC, 


9. A trapezoid as 
shown in the accom- 
panying diagram. 


10. A trapezium one of whose nor ai 
diagonals measures 42 yards, the 
perpendiculars to the opposite cor- 
ners measuring 18 yd. and 17 yd., 
respectively. 


CHAPTER X. 
ALGEBRAIC EQUATIONS. 


ONE UNKNOWN QUANTITY. 


848. Oral Exercises. 

1. What number increased by 12 equals 16? 
9 added to a number equals 14. Find the number. 
What number diminished by 7 equals 8? 
18 diminished by what number has 10 for remainder? 
Hight times what number equals 64? 
What number multiplied by 9 gives 63 for product? 


Sas ee ee ee 


Three times a certain number added to twice the same 
fmber equals 40. What is the number? 


8. 36 is equal to 10 times what number added to 8 times the 
same number? 


9. The difference between 4 times a number and 38 times the 
3ame number is 72. What is the number? 


10. Twice a number subtracted from 11 times the same num- 
ber equals 27. What is the number? 


— 849. Sight Exercises. 


Give values of x, y. 2, etc - 


1. ?+12=16 4. 182—?=10 
2. 9+°9=14 5. 82= 64 
3. ?—T=8 6. 9y=63 


369 


370 ARITHMETIC. 


7 32+2x2%=40 12. 10y+8y-—-4y = 42 
8. 102+82—86 13; (De 4 4s Bo de 
ody = By = 2 1a) 8244230280 

10. llw—2w=27 15. 12y—5y=25410 
i. Sp Ox ba bd 16. 6w+60=16--8 


850. Slate Problems. 


1. A horse and a wagon cost together $600. What is the 
price of each, if the wagon costs twice as much as the horse? 


Let x = cost of horse; 
then 2x = cost of wagon. 
Cost of both = 24% +2=600 
32 = 600 
= 200 
2a = 400 


Ans. Cost of horse, $200; of wagon, $400 


2. Divide 100 into two parts, one of which shall be four times 
as large as the other. 


Let x” = one part ; 
then 4a = the other. 
x+42x= 100. 


3. $18,000 is divided among three children, the second of 
whom receives twice as much as the first, and the third of whom 
receives six times as much as the first. Required the share of 
each. 

v, 22, 62. 

4. In a class of 54 pupils, there are twice as many boys as 

girls. How many are there of each? 


5. The sum of two numbers is 78. One is five times as large 
as the other. What are the numbers? 


6. 156 is equal to seven times a number added to five times 
the same number. Find the number. 


ALGEBRAIC EQUATIONS. ole 


7. The difference between three times a certain number and 
nine times the same number is 66. What is the number? 


8. $27,000 is divided among three children, the second of 
whom receives twice as much as the first, and the third of whom 


receives three times as much as the second. What is the share 
of each? 


9. The sum of two numbers is 72, and the greater is 5 times 
the other. What are the numbers? 


10. John, Henry, and James have 54 marbles. Henry has 
twice as many as John, and James has as many as the other 
two. How many has each? 


11. The sum of the ages of mother and daughter is 42 years. 
What is the age of each, if the mother’s age is six times that of 
her daughter ? 


12. A man paid $96 for an equal number of hats and coats, 
paying $2 apiece for the former and $10 apiece for the latter. 
How many of each did he buy ? 


13. Divide 41 into four parts, the first being twice the second, 
the second three times the third, and the third four times the 


fourth. 
(Let « = the fourth.) 


14. The sum of three numbers is 180. The first is double the 
second, and the third is three times as large as the sum of the 
other two. What are the numbers? 


15. Mr. Smith paid 81 cents for sugar and flour, the same 
quantity of each. For the sugar he gave 5¢ per pound, and for 
the flour 4% per pound. How many pounds of each did he buy? 


16. The length of a rectangular field is 24 rods, its breadth is 
x rods, its area is 456 square rods. Find the value of z. 


17. It takes 340 feet of fence to enclose a square lot. What 
are the dimensions of the lot? 


B12 ARITHMETIC. 


18. Mrs. B. divides $120 between her son and her daughter. 
She gives the latter twice as much as she gives the former. 
What is the share of each? 


19. The earnings of a man and his son during January 
amounted to $175, both having worked the same number of 
days. The father’s wages were $4 per day, and the son’s wages 
were $3 per day. How many days did they work? 


20. The sum of $240 is divided among four children, two 
boys and two giris. Find the share of each, if each girl’s share 
is double that of each boy. 


21. A man worked twice as many days as his son. Their 
combined earnings amounted to $165. Find the number of days 
each worked, if the father earned $4 per day and the son three- 
fourths as much per day. 


22. A boy’s bank contains 78% in dimes, nickels, and cents. 
There are twice as many nickels as there are dimes, and three 


times as many cents as there are nickels. How many are there © 
of each? 


23. I paid 75¢ more for a roll of 15-cent ribbon than I did 
for a roll of 12-cent ribbon of the same length. How many 
yards did each roll contain? 


24. A rectangular field whose length is four times its breadth 
requires 250 rods of fence to enclose it. What are the dimensions 
of the field? (Make diagram.) 


25. A girl paid 60 cents for a speller and a reader, the cost 
of the former being one-third that of the latter. Find the cost 
of each. 


26. The sum of two numbers is 72, and the smaller is one- 
fifth of the other. What are the numbers? 


27. Mary, Susan, and Jane have 54 hickory nuts. Susan has 
one-half as many as Mary, and Jane has as many as the other 
two. How many has each? 


ALGEBRAIC EQUATIONS. ote 


CLEARING OF FRACTIONS. 
851. Oral Exercises. 
. One-fifth of a number is 4. What is the number? 
- +ofanumber is 8. What is 2 of the number? 


1 

2 

38. 4+ ofa number is 12. What is the number? 

4. +ofanumber is 10. What is $ of the number? 
5 


. If? ofa number is 30, what is the number? 


6. One-half a number added to + of the same number equals 
what fraction of the number ? 


7. One-half a number added to + of the same number equals 
30. What is the number? 


8. One-third of a number + one-sixth of the number = what 
fraction of the number? 


9. One-third of a number added to + of the number = what 
fraction of the number? 


10. 42-+42 = what fraction of x? al Tee 


852. When x = 82, find the value of three-fourths of 2; 
OL 
TAG 
When 2, (3x2 divided by 4) = 24, what is the value of 
Arie roe a 
Find the value of y, when ‘= 12. Of 2y, when = = 24, 
2) 


1.€., 


Given the equation “3 = 20; by what whole number can we 


multiply the first member to get rid of the fraction? If we mul- 
tiply one member of an equation by any number, what must we 
do to the second member in order to preserve the equality ? 


37 


4 


853. Sight Exercises. 


Give values of z, y, z, etc. : 


be 


say 5. 
5 
eels: 6 
. 
z 
oy 7 
4 
3¥ 9] 8 
4 


854. Slate Exercises. 


1. 


Find the value of the unknown quantity (2). 


ARITHMETIC. 


ena. 
eit 
ete 


a 


— 
= 


18 be 


12. 


In each of the following equations, multiply both members by the least 
common denominator of the fractions. 


xv x x 
ee OG 
a3 14 


Multiplying by 12, we have 6a+42+432= 312 


eT 
—-+—= 85 
3° 3 
eee 
“= 49 10. 
314 
eee es 
; Hah aa wet 11. 
S¢e+2x2=92 12 
pie Ea O99) 13", 
3 4 


9. 


2h2=115 14. 


me ie te 
aye 

a2 —“_—= 156 
32 

OF — OY 

2 

lie = 27 
lz _ 99 

4 
98 = 44 


15. 


16. 


17. 


18. 


19. 


20. 


38x 
2 pate tet 
a+ 4 
x 
=e 
idadirs 
15% 882 
10093560 
dba — 28¢4=45 
Seen 
ee OG 
WS ine 
33 2=116 


ALGEBRAIC EQUATIONS. 375 


“Be ay eae qe nate 
OP Sa = OD DE: Nigh eed lS cetiacr S eh 9) 
o13t4 9 ar 3) 2 
Lf Sa 
aes Speen ae St GF 5. iit == '80 
x Shae T 25. x ‘1 


7 Ee Ae aS 2 32x 
Foie eee a aes —“ = 94 
2 5 9 “+ n 26. ©+2a4+ 7 2 


855. Slate Problems. 


1. The sum of two numbers is 90, and the smaller number is 
one-fifth of the larger one. What are the numbers? 


Hh 
q — = 90, 
(#45 ) 


2. Divide 100 into two parts, one of which shall be 24 times 
the other. 
3. After losing + of his money, a man has $714. How many 
dollars had he at first? 
(« ~2- 714.) 


4. A horse was sold for $240, the seller thereby gaining one- 
third of what he originally paid for it. How much did he pay 


for it? 
x 


5. One-half of a number added to one-fourth of the same 
number equals 663. What is the number? 

6. The difference between 3 of a number and $ of the same 
number is 15. Find the number. 

7. One number is 2 of another. Their sum is 55. What 
are the numbers? 

8. Find a fraction equivalent to 4, the sum of its numerator 
and its denominator being 60. 

(Let 7” = numerator and 8x = denominator.) 

9. Find a fraction equivalent to #, the difference between its 

numerator and its denominator being 24. 


376 ARITHMETIC. 


10. The sum of two numbers is 480, and the quotient obtained 
by dividing the greater by the less is 7. What are the numbers? 


11. Find two numbers whose difference is 522 and whose 
quotient is 30. 


12. A boy buys apples at 2¢, pears at 3, and oranges at 4f, 
the same number of each. How many of each does he buy, if 
he pays 81¢ for all? 


13. A girl bought 70 cents’ worth of peaches and plums. 
She paid 8¢ each for the peaches and 2¢ each for the plums, 
buying four times as many of the former as of the latter. How 
many of each did she buy? 


14. $1,500 is divided among three persons, the second of 
whom receives three times as much as the first, and the third 
three and one-half times as much as the first. Find the share 
of each. 


15. A farmer paid for a cow three-sevenths as much as he 
paid for a horse. How much did he pay for each, if the latter 
cost $80 more than the former? 


16. Three times a man’s money increased by two-thirds of his 
money is equal to $1,100. How much money has he? 


17. After giving away 2 of his marbles and losing 4 of them, 
Joseph has 24 left. How many had he at first? 


18. Bought a coat, a hat, and an umbrella for $15, paying 
for the hat 11 times as much as for the umbrella, and for the 
coat 34 times as much as for the hat. Find the price of each. 


19. A merchant purchased two pieces of cloth for $240, pay- 
ing for one piece twice as much per yard as for the other. The 
former contains 36 yards and the latter 48 yards. How much 
does he pay per yard for each? 


20. A farmer sold 4 times as many cows as horses, receiving 
for all $840, at the rate of $40 for a cow and $120 for a horse. 
How many of each did he sell? 


ALGEBRAIC EQUATIONS. ate. 


TRA NSPOSING. 
856. Sight Exercises. 


Give values of x, y, z, etc.: 


1. o+15=21 % 8y+6=15 
2. 2y+15=21 «8. Ty—138=15 
3. 2-7=21 9. 9y+13=58 
4. 4w—T=21 10. 3y—10= 56 
30 
he oe The eee 
5. 5 +3 8 a 7 
4w 
Fi (Go a 122. —4@—J=11 
2 o 


857. lf 2z+15= 21, x= 21 — what? 

When x — 7= 21, x = 21-+ what? 

If in the equation 22-++15—=21, we take away 15 from the 
first member, what must we do to the second member to preserve 
the equality ? 

By transposing we mean bringing the unknown quantities 


(x, y, 2, etc.) to one side of the equation, and the known quan- 
tities to the other. 


Notr. — In bringing a quantity from one side of the equation to the other 
the sign of the quantity is changed. 


858. Slate Exercises. 


Find values of the unknown quantities. 


Note. — Clear of fractions when necessary ; then transpose. 


Dat ba DO 5. e+387=—25+4+11 
Bee 03 pa eed 

3. 82 —48=98 Mee OO lee 8 

EEE ane SMe yard 


378 ARITHMETIC. 


9. 82—6=48+2 15. Tz—5x2=20+2+4+4 
10. 87+6=9-—-227412 16. 6a4—14=16+2 
11, 2a—2—16=2+10 17. 2x—11+62—60=524+25 
x gr 
12. ——8= 24 . ~+=-—5=10 
3 8 18 313 
x ka hier 
13. —-+4—7=21 " 19. 22—6=16+=—= 
6 if LAY dis 
oY iar SP lee ie tay: 
. per is . 2 eg eee Ae 
14, F4+5=10+5 20. Qe po Fae 


859. Slate Problems. 


1. The sum of three numbersis 51. The second is 5 less than 
the first, and the third is 10 less than the first. What are the 


numbers ? 
Let x = first number, 
x —5 = second number, 
x — 10 = third number; 
x+x2—54+2—10=51. 
Transposing - et+ae+e=51+45+10, 


Sa'= 66; 
x = 22, first number, 
x —5= 17, second number, 
x — 10= 12, third number. 


2. Add 45 to four times a number, and you will have seven 

times that number. What is the number? 
(7#= 45 + 42.) 

3. Nine times a number less 27 equals six times the number. 
Find the number. 

4. Two boys have together 48 marbles. One has 18 more 
than the other. How many has each? 

(x, « + 18.) 


ALGEBRAIC EQUATIONS. ; 379 


5. The length of a rectangular lot is 75 feet more than the 
breadth. The distance around it is 250 feet. What are its 
dimensions? 


6. A piece of land containing 86 acres is to be divided into 
two fields, one of which shall be 8 acres larger than the other. 
How many acres in each field? 

7. Ata certain election 2.436 votes were cast for two can- 


didates, the successful one receiving 318 more votes than his 
opponent. How many votes did each receive? 


8. A man, being asked his age, replied that if he were half 
as old again and 7 years more he would be 100. What was his 
age? 

9. The sum of two numbers is 96, and their difference is 72. 
Find the numbers. 


(Let x = less, x + 72 = greater.) 


10. After paying 4 and 4 of my debts, I still owe $45. How 
much did I owe originally? 


11. Divide 45 into two parts, one of which shall be 6 less 
than twice the other. 


12. William has $5 more than John, and three times William’s 
money added to five times John’s would be $1038. How many 
dollars has each? 


13. I bought 3 cows and 4 horses for $635, paying $80 apiece 
less for the cows than for the horses. How many dollars apiece 
did I pay for each? 


14. Mary has a dollar in dimes and five-cent pieces. She 
has 11 more of the latter than of the former. Find the number 
of pieces of each denomination. 


15. Divide 100 into two parts whose difference shall be 48. 


16. Ina class of 54 pupils, the girls outnumber the boys by 
12. How many are there of each? 


380 ARITHMETIC. 


17. $18,000 is divided among three children, the second of 
whom receives $2,400 more than the first, and the third of whom 
receives $2,400 more than the second. Find the share of each. 


18. The greater of two numbers is 11 more than 3 times the 
less. Their difference is 838. What are the numbers? 


19. A boy spent a dollar for postal cards, 2-cent stamps, and 
5-cent stamps. He bought 15 more 2-cent stamps than 5-cent 
stamps, and 15 more postal cards than 2-cent stamps. How 
many of each did he buy? 


20. A farmer has 88 head of stock — horses, cows, and sheep. 
He has 17 more cows than horses, and the number of sheep is 24 
greater than that of the cows and horses together. How many 
are there of each? 


CHAPTER XI. 


PERCENTAGE. — INTEREST. — DISCOUNT. — SURFACES AND 
VOLUMES. 


PERCENTAGE. 


860. Preliminary Exercises. 


Per cent means hundredths. Seven per cent means seven 
hundredths, ;75, or .07. It is written 7%. 


How many hundredths of a number is its half? +—how 
many hundredths? 4? 7)? &? wy? 2° 2? 


What per cent of a number is the half of it? 1? 4? 4% 


2 41 
6° 
1 1 1 1 1 I 

we? eo Bo tel gr? te? oe? ae? ae? to? abe? oh! 


3 5 
Ferre eke roe Te! ty sot we eee Coe 


862. 1 per cent of a number is equal to what fraction of it? 
2%? 3%? 4%? 5%? 6%? TH? 8H? 9%? 10%? 12%? 
15%? 20%? 25%? 380%? 386%? 40%? 50%? 60%? 
75%? 80%? 90%? 


863. What fractions are equal to the following: 
124%? 162%? 884%? 87h? G49? 62957? 3496? 
625%? 663 iat 875%? F%? 4%? BAGH? 1%? 


864. 3 times a number is what per cent of it? 24 times? 11 
times? 164 times? 44 times? 9; times? 
381 


382 


ARITHMETIC, 


865. Oral Exercises. 


Ls 


10. 


Co AFT Pw N 


Find 123% of 1 gal. 


374% of $24 
334% of 81 cows 
6% of 150 lb. 
4% of 125 bu. 
621% of 1 pk. 
AL of $200 
99% of 200 gal. 
4% of $640 

1%, of 800 yd. 


866. Slate Eeenien 


ie 


ee ee ee 
Pm wo WDM FS OS 


Soe Corea eraee Serer oo 7 gee 


Find 62% of $95.10 


12% of $37.50 
331% of $28.80 
1% of $1,240 
4L% of $92.40 
450% of $92.40 
20% of $51.60 
1,400% of $89.70 
124% of $73.28 
131% of $27.60 


. 614% of $25.60 
. 81% of $47.40 
. 51% of $29.50 
. 860% of $38 


. 662% of 66 horses 
. 162% of l yd. 

- 81% of $300 

. 24% of 80 sheep 

. 40% of $2.50 

- 20% of 65 rd. 

- 10% of 15 lb. 

. 34% of $60 

- £% of $72 

. 14% of $96 


. 4% of $24 

. 25% of $52.36 
- 60% of $33.30 
. 6% of $19.50 
. 62% of $47.40 
. 12% of $62.50 
- 44% of $71.50 
- 40% of $28.30 
. 160% of $39.40 
. 84% of $23.75 
. 662% of $825 
. 75% of $59.20 
. 821% of $392 
. 933% of $4.96 


PERCENTAGE. 383 


TO FIND THE BASE OR THE RATE. 


867. Slate Problems. 
29. Find 25% of 280. 


280 is called the dase, 25 is called ne rate. 


Multiplying 280 by ;%%5, we get 70, which is called the per- 
centage; that is, 


Rate 
100 


30. Find z per cent of 65. 

31. If x per cent of 65 is 26, find the value of 2. 

32. Find 25% of z. 

33. If 25 per cent of x is 42, what is the value of 2? 
34. Increase x by 20% of itself. 


868. Fase < 


= Percentage. 


35. If « increased by 20 per cent of x equals 132, what is the 
value of x? 


36. Diminish x by 334% of itself. 


37. Find the value of x, when x diminished by 334 per cent 
of itself equals 78. 


38. Find 7% of 2. 

39. What is the value of x, if 7% of 2 = 34? 

40. Find 4% of x. 

41. Find the value of x, when 1% of x = 28. 

42. What (x) per cent of 65 is 26? 

43. 24 is 18 per cent of what number (x)? 

44. 250% of what number (x) = 180? 

45. What number (z) increased by 25% of itself equals 85? 


46. $= what per cent of 4? 


384 ARITHMETIC. 


47. 1s what per cent of 4? 


<<. of 2; ie, 5. —+. Clear of fractions and solve. 

48. Find the interest on x atahies for 1 year at 54%. 

49. What sum at 51% gives a yearly interest of $44? 

50. What is 662% of 2? 

51. What per cent of 88 is 33? 

52. Find the difference between 2 of 800 and 4 per cent of 800. 


53. Find the selling price of a horse that cost $175 and was 
sold at 25% profit. 


54. How much insurance does a man receive for $12.50 when 
the rate is 24%? 


55. What is 16% of 64? 

56. 311s what per cent of 2? 

57. What per cent of $389.50 is $124.64? 

58. $174.04 is 99% of what sum of money? 

59. What number increased by 16% of itself equals 1,276? 
60. 984 is 1834% of what number? 


61. 2% of a number is 81. What is the number? 
3 


62. An importer paid duties amounting to $386.75. If the 
duty was 25% of the value of the goods, what was their value? 


63. A collector deducts 24% commission, and returns to his 
employer $745.68. How much did he collect? 


pea eer 7db GS: or. e — 5 745.68; or, 22% — 745.68. 
100 40 40 


PERCENTAGE. 385 


64. A commission merchant receives 24% commission for buy- 
ing grain for a customer. The cost of the grain and his commis: 
sion amount to $4,223. How much does the grain cost? 


x = cost of the grain ; rie commission. 


65. A capitalist sends a commission merchant $8,670 to invest 
in cotton and to include commission at 2%. How much does 
the commission amount to? 


66. A commission merchant receives $1,071 to invest in oats 
after deducting 2% commission. How many bushels of oats at 
30 per bushel does he purchase? 


Should the commission merchant deduct 2% of $1,071, or 2% of the cost 
of the oats? 


67. A house is insured for 2 of its value at 7%. The annual 
cost (premium) is $8.40. What is the value of the house? 


Let = value. Then 2a gn eae ee 

3 800 1200 
68. What will be the taxes on a house worth $48,000 and 
assessed at 2 of its value, the tax rate being $18.50 per $1,000 


of assessed value? 


869. Oral Exercises. 
1. 3 is what part of 6? 
3 1s what decimal of 6? 
3 1s how many hundredths of 6? 
3 is what per cent of 6? 
6 is what per cent of 3? 
What number is 50% of 6? 
3 is 50% of what number? 
2is what % of 100? 
21s what % of 200? 


CO AH TP wo LW 


386 ARITHMETIC, 


10. What number is 5% of 100? 

11. What % of 20 is 1? 

12. 4is what % of 200? 

13. 3is 4% of what number? 

14. What per cent of 9 is 201? 

15. What number, increased by 1 of itself, equals 10? 
16. What number, increased by 25% of itself, equals 20? 
17. 65 diminished by 5% of itself, equals what ? 

18. Buying price $100, selling price $112.50. Gain %? 
19. Cost $80, profit 20%. Selling price? 

20. What principal will give $30 yearly interest at 6%? 


21. At + of 1%, how much will I pay for insurance on 


$ 10,000 ? 
22. If insurance costs 3%, how much can I get for $60? 
23. Bought sugar at 4% per lb.; sold it for 5f. Profit %? 
Norr. — Per cent of profit is based upon the cost. 
24. 21s what % of 2? 
(Reduce both to a common denominator.) 


25. 1d is what % of 62? 


PROFIT AND LOSS. 
870. Slate Problems. 
Find the profit or loss, and the selling price: 
1. Cost $1,876; gain 15%. 
2. Cost $36.75; loss 20%. 
3. Cost $1,012.50; gain 162%. 
4. Cost $875; loss 5%. 
5. Cost $934.56; gain 123%. 


PROFIT AND LOSS. 387 


Find the profit or loss per cent. 


N. B. — Profit or loss per cent is based upon the cost. 


6. 


Cost $600; selling price $618. 
Gain or loss = x% of cost. 18 = x% of 600. 


. Cost $1,903; selling price $802. 


Cost $86.20; selling price $73.27. 


. Cost $908.40; selling price $1,090.08. 
. Cost $84; selling price $78.75. 
. Selling price $78.75; loss $5.25. 


Gain or loss = x of cost. 


. Selling price $150; gain $25, 

. Selling price $831.25; loss $438.75. 

. Selling price $1,051.88; gain $116.82. 
. Selling price $8438.75; loss $168.75. 


Find the cost, and the profit or loss: 


16. 


i A 
Ss 
19; 
20. 
21. 


Selling price $468.75; gain 25%. 
Let x = cost. 
Selling price $73.84; loss 20%. 
Selling price $1,646.08; gain 334%. 
Selling price $204; loss 15%. 
Selling price $66.30; gain 4%. 
A man buys a horse for $275, and sells it at a profit of 


20 per cent. How much does he gain? 


22. 


A cow is sold for $75, on which the profit is $15. What 


is the gain per cent? 


23. 


cost. 


A lot is sold for $960, which is 20 per cent more than it 
Find the cost of the lot. 


388 ARITHMETIC. 


24. Tea that costs 82% per pound is sold for 48¢. What is 


the gain per cent? 


25. A man buysa horse for $175 and sells it for $200. What 
per cent does he gain? 


26. What per cent is lost on a horse costing $200, and sold at 
a loss of $25? 


27. What is the selling price of dress goods costing 331 ¢ per 
yard, on which a profit of 124 per cent is made? 


28. Sold a coat for $33.60, thereby losing 16 per cent. What 
was its cost ? 


29. How much did I gain ona house for which I paid $8,760, 
my profit being 24 per cent? 


30. A man after spending 10 per cent of his money for 
clothing, 25 per cent for board, and 30 per cent for incidentals, 
has $70 left. What did he have at first? 


31. The population of a city was 16,000 in 1880. In 1890 it 
was 22,000. What was the gain per cent in ten years? 
(Which is the base ?) | 
32. The population of a city was 30,000 in 1890, a gain of 


6,000 over the previous census. What was the increase per cent ? 


(Have the correct base.) 


33. 16 shares of bank stock, face (par) value $100 each, were 
sold at 2% per cent above the face value. How much was 
received for the stock ? 


34. What will be the cost of 84 shares of stock, face (par) 
value $50 each, at 31 per cent below par? 


35. Find the interest on $784.50, at 4 per cent, for 3 years 
7 months 15 days. 


SURFACES AND VOLUMES. 389 


MEASUREMENTS. 


871. Slate Problems. 


1. A tank 18 feet long, 15 feet wide, contains 30 cubic yards. 
What is its depth in feet? 


ro 


18 «x 15 = 30 27: «= 
( x Xz x e 18x 15 


2. The surface of the walls and the ceiling of a room 24 feet 
long and 12 feet high is 1,440 square feet. What is the width 


of the room? 
288 + 12a + 288 + 122 + 24a = 1,440. 


122 sq. ft. 288 sq. ft. 


24 ft. x ft. 24 ft. x ft. 


3. A man owns a rectangular plot of ground 132 feet long, 
110 feet wide. He divides it into two equal parts by a fence 
running from the north-east to the south-west corner. What 
part of an acre does each piece contain ? 


Indicate operations. Cancel. 


4. How many yards of fence will be needed to enclose a 
rectangular field, 80 rods long, containing 25 acres? 


5. Make a diagram of a floor 6 yards long, 4 yards wide. 
Show how many strips of carpet, 27 inches wide, running cross- 
wise, will be needed to carpet the room, no allowance being made 
for waste? 


390 ARITHMETIC. 


6. How many sheets of tin 6 inches by 4 inches will be 
required to cover a roof 20 feet by 30 feet, making no allowance 
for overlapping? 


7. The owner of a lot 25 feet by 100 wishes to build around 
it a tight board fence 6 feet high. How many running feet of 
fence will there be? How many square feet? | 


(Draw the ‘‘ development” of the fence, marking on it the dimensions.) 


8. How many boards 12 feet long, 6 inches wide, will it take 
to build such a fence? 


9. A plot of ground 300 feet by 200 feet is bought at $181.50 
per acre. It is sold in lots 100 feet square at $160 each. What 
is the profit ? 

10. How many cakes of ice 4 feet long, 2 feet wide, can be cut 
from a rectangular pond 320 feet long, 160 feet wide, no allowance 
being made for waste? (Cancel.) 


11. If the ice is 14 feet thick, how many cubic feet of ice will 
there be? Find its weight in tons, at 57.5 pounds per cubic foot. 


12. Give the length of a shed 15 feet high, 32 feet wide, that 
will exactly hold the ice cut from the above pond. (Cancel.) 


13. The owner of a plot of ground 640 feet long, 440 feet 
wide, cuts two streets, each 40 feet wide, through the middle of 
the plot, one running north and south, the other east and west. 
How many square feet of land had he originally? How many 
square feet has he left for building purposes? (Make diagram.) 


14. What will it cost him, at 60% per square yard, to make 
the above streets? 


15. Ifthe remaining ground is divided into building lots 25 
by 100, how many lots will there be? 


16. How many square yards of plastering will be required 
for the walls and the ceiling of a room 21 feet wide, 30 feet long, 
9 feet high, deducting for 2 windows, each 6 feet by 3 feet, and 
a door 8 feet by 44 feet? 


PERCENTAGE. 


PROFIT AND LOSS. 


872. Slate Exercises. 


1. 


SO AH TP w W 


Pe <~ ne <~ er << > 
Pond FH FSO OD UWA eT PF WY DW HO 


Bought for $36; sold for $40. Gain per cent? 


Bought for $40; sold for $36. Loss per cent? 
Cost 36¢; selling price 40%. Gain per cent? 
Cost $24; gain 10%. Selling price? 

Selling price 70%; loss 75%. Cost? 

Buying price 70%; gain 75%. Selling price? 
Cost $20; selling price $29. Gain %? 

Cost $20; selling price $290. Gain %? 

Cost $20; selling price $20.90. Gain %? 
Cost $20; selling price $20.09. Gain %? 


. Selling price $300; loss $100. Loss %? 

- Cost $300; gain $100. Gain %?°* 

. Selling price $175; cost $150. Gain %? 
. Selling price $375; gain 25%. Profit? 

. Cost $36.50; selling price $28.50. Loss % ? 
. Selling price $33.95; loss 3%. Cost? 

. Cost $75.50; loss 54%. Selling price? 

. Selling price $20.16; gain 5%. Cost? 

. Selling price $64; profit $16. Gain %? 
. Cost $37.50; selling price $42. Gain %? 
. Selling price $26.88; loss 62%. Loss? 

. Cost $4. gain 1%. Selling price? 

. Selling price $41.16; gain5%. Cost? 

. Selling price $29.83; loss 5%. Loss? 


391 


392 ARITHMETIC. 


25. Cost $19.50; loss 6%. Selling price? 

26. Cost $84; selling price $184. Gain %? 

27. Selling price $700; gain 250%. Cost? 

28. Cost $324.80; gain 175%. Selling price? 
29. Selling price $6.50; loss 133%. Loss? 

30. Cost $346.50; selling price $339.57. Loss %? 
31. Selling price $17.64; loss 2%. Cost? 

32. Cost $4,613; gain 134%. Profit? 

33. Selling price $26.69; gain 61%. Profit? 

34. Cost $8,766; loss 7%. Loss? 

35. Selling price $50.00; profit $4.20. Gain %? 
36. Cost $37.50; gain $5.40. Gain %? 

37. Selling price $205.20; loss $45.90. Loss %? 
38. Cost $25.60; gain 64%. Selling price? 

39. Selling price $17.35; loss $%. Loss? 

40. Profit $36; gain4%. Cost? 

41. Loss $28.17; loss 3;3,%. Selling price? 
42. Cost $3,864.25; loss 8%. Loss? } 
43. Selling price $89.37; profit $6.62. Gain %? 
44. Cost $82; loss $2. Loss %? 

45. Selling price $22.85; gain 62%. Cost? 

46. Profit $47.25; gain 74%. Selling price? 

47. Loss $38.46; loss 4%. Cost? 

48. Cost $75.52; gain 831%. Profit? 

49. Selling price $49.95; loss $4.05. Loss %? 
50. Cost $3,879; loss 8%. Loss? 


PERCENTAGE. 393 


INTEREST. 


873. In calculating interest, take 30 days to a month, 12 
months to a year. 


874, Principal x fe 


x Time (wn years) = Interest. 


100 
Notre. — Change given time to years. 
2 yr. 6 mo. =2)hyr. =$ yr. 
1 yr. 7 mo. =19mo. =1$ yr. 
4mo.10da.=41} mo. = 3 yr. = yr. 
1734 


lyr. 5 mo. 15 da. = 17} mo. = — yr 
5 mo. 17 da. = 167 da. = 487 yr. 


875. Slate Exercises. 
Find the ipiareat on: 
1. $750, for 24 years, at 6%. 
$ 84.75, 34 months, at 4%. 
. $308.25, from Oct. 1 to Oct. 21, at 5%. 
$464.75, 8 mo. 12 da., at 6%. 
. $360, 33 da., at 5%. 
$94.43, 2 mo. 5 da., at 7%. 
$400, 1 yr. 1 mo. 1 da., at 43%. 
$720, 21 da., 7%. 
$1,000, 8 da., 5%. 
$630, from April 1, 1890, to Jan. 16, 1892, at 6%. 


1892 —1—16 
1890 —4— 1 


11. $394.50, 2 yr. 1 mo. 7 da., at 6%. 
12. $1,560, 3 yr. 4 mo. 9 da., at 44%. 


OO AFT P w DN 


_ 
2 


394 ARITHMETIC. 


13. $960, 11 mo. 24 da., 2%. 
14. $86.40, 1 yr. 9 mo. 20 da., 5%. 
15. $108.36, 4 yr. 7 mo. 10 da., 33%. 


876. Amount = Principal + Interest. 

Find the amount: 

16. $813, from April 19, 1889, to March 4, 1894, at 6%. 
17. $960, from Jan 1, 1898, to Dec. 21, 1894, at 4%. 
18. $27.84, for 3 yr. 6 mo. 9 da., at 6%. 

19. $48.90, for 17 da., at 6%. 

20. $144, for 2 yr. 5 da., at 32%. 

21. $834.76, for 15 mo. 27 da., at 44%. 

22. $5,760, for 1 yr. 5 mo. 29 da., at 5%. 

23. $2,346.50, for 7 yr. 13 da., at 3%. 

24. $1,892, for 3 yr. 5 mo., at 7%. 

25. $150.40, for 1 yr. 2 mo. 3 da., at 6%. 


_ 877. Interest-bearing Demand Notes. 

26. San Francisco, Jan. 7, 1893. 

On demand, I promise to pay William Britt, or order, Seven 
Hundred Sixty-five 549% Dollars, value received, with interest 
at 6 per cent. 

$ 765545. ARTHUR TOWNSEND. 


How much money will be required to pay the above note, 
with interest, July 15, 1894? 

27. A demand note, dated Sept. 25, 1892, with interest at 
8% from date, is paid Jan. 2, 1895. How much was due, the 
face of the note being $750? : 

28. Find the amount due March 4, 1894, on a note for 
$365.84, dated May 20, 1892, with interest from date at 7%. 


INTEREST. 395 


29. Find the amount necessary, Oct. 16, 1896, to pay a note 
of $1,240, with interest at 6% from Aug. 15, 1892. 


30. An interest-bearing note for $87.60 is dated April 38, 
1886. How much is due on it for principal and interest Jan. 2, 
1894? Rate 44%. 

878. Oral Problems. 

1. Find the interest on $300, for 1 year 7 months, at 4%. 
$12 per year is how much for 7 months? 

2. On $60, for 33 days, at 6%. 
$3.60 for 360 days is how much for 33 days? 

3. On $120, from Jan. 1, 18938, to July 1, 1894, at 5%. 


4. How long will it take $100 to produce $15 interest at 
6%? 
5. At what rate per cent will $50 produce $6 in 2 years? 


6. What is the interest on $300, at 6%, from Feb. 1 to 
Feb. 21? 


7. What part of a year is 72 days? 

8. Find the interest at 4%, for 90 days, on $150. 

9. On $240, for 36 days, at 5%. 
10. What is the amount of $200, for 3 years 1 month, at 6%? 
11. How long will it take $1 to make $1 interest at 5% ? 
12. How long will it take any sum to double itself at 6% ? 
13. How long will it take $14.90 to double itself at 4% ? 


14. At per cent per month, find the interest on $90 for 16 
months. 


15. 5% per year is 1% for how many days? 
16. 41% per year is 1% for how many days? 
17. Find the interest on $75, at 5%, for 72 days. 


396 ARITHMETIC. 


SPECIAL DRILLS. 
884. Give sums: 


495 + 99 99914295 $2634+$6.37  $5.45+ $9.99 
99+576 5764999 $4564+$2.84  $9.99+ $6.78 
685+ 99 999+685 $649+$3.12 §$12.68+ $0.99 
99 +599 599+999 $3.58+$5.67  $0.99+$13.33 
885. Give differences : 
565-— 99 1424-999 $7.00-$2.63 $15.44— $9.99 
488— 99 1,575—999 $640—$8.56 $9.44 — $6.45 
794— 99 1,684—999 $9.61—$4.49 $7.88 -— $4.89 
898— 99 1,598—999 $815 —$5.58 $9.53 — $2.99 
886. Give products: 
24 x 21 21 x 31 41 x 41 19 x 374 
03 X 21 02 X 31 32 x 41 18 x 624 
42 x 21 43 x 31 21 x 41 22 X 662 
51 Xx 21 dl x 31 42 x 41 33 xX 75 
887. Give quotients: 
Poe 25 228 +19 9A + 331 656 + 16 
18 + .25 234 +18 36 + .334 544-17 
24 + .75 238 +17 16 + .662 558 + 18 
36 + .75 336 + 16 66 + 662 418+19 
888. Give answers: 
147 x 7 1334 x 8 13 x 134 20 x 82 
152 x 6 212 x 4 14 x 12% 21 x 73 
55 + 5 Ti + 3 15 + 3h 63 + 34 
142-124 63-8 20 ++ 88 64 + 54 


REVIEW. 397 


889. Oral Problems. 

1. What will be the cost of 48 yards of cloth at 871 per 
yard? 

2. How many square yards in a piece of carpet 48 yards long, 
27 inches wide? 

3. How many yards of carpet 27 inches wide will be needed 
to cover a floor containing 48 square yards? 

4. Paid $3.45 for groceries, $1.50 for dry goods, and 99% 
for sundries. What is the total? 

5. From a chest containing 254 pounds of tea, 84 pounds 
were sold. How many pounds remain? 

6. At 8719 per peck, what will I receive for 4 bushels of 
potatoes ? 


7. 831 yards of cloth are divided into 9 pieces. How many 
yards are there in each piece? 


8. I buy hardware to the amount of $6.37. I give the store- 
keeper two $5-bills. How much change should I receive? 
9. What will be the cost of 24 yards of calico at 42¢ per yd.? 
10. What will I have to pay for 19 base-balls at $1.25 each ? 
11. At $1.874 per yard, what will be the cost of 120 yards of 
silk ? 
12. For $120, how many yards of silk can I buy at $1.873 
per yard? 
13. What will be the cost of a ton of hay at 971 per ewt.? 
14. A square field requires 320 rods of fence. How many 
square rods are there in the field? 
15. How many acres are 6,400 sq. rd.? 
16. At 48¢ per yd., how many yards of calico can I buy for 
95 ¢? 
17. If slate pencils oost 2 mills each, how many will I receive 
for $4? © ‘ 


398 ARITHMETIC. 


18. At $5.00 per ton, how many pounds of coal can be bought 
for 1¢? 
19. Find the cost of 8 T. 480 lb. coal at $5 per ton. 


20. At $5 per ton, how many tons and pounds of coal can I 
buy for $10.80? 

21. How many square yards are there in a field 41 yards 
Wng, 42 yards wide? 

22. If I pay 15¢ for 34 yards of muslin, what is the price per 
yard? 

23. How many acres of land are there in two farms contain- 
ing, respectively, 347 and 495 acres? 

24. At871¢ each, how many base-balls can be bought for $56? 


25. If one man can do a piece of work in 24 days, and an- 
other man can do it in 48 days, how long will it take both, 
working together ? 


APPROXIMATIONS. 


890. Give approximate answers at sight (Art. 521): 


1. Find the interest of $150, at 4%, from Jan. 1, 1893, to 
Dec. 30, 1895. (Nearly 3 years.) 


2. What is the weight, at 574 lb. per cu. ft., of a cake of ice 
4 ft. by 2 ft. by 14 ft.? (Nearly 60 lb. per cu. ft.) 


3. Find the amount of goods sold, the commission at 27% 
being $11.75. (About 3%.) 
4. What % of 497 is 249? 
5. What % of 31% is 1134? 
6. Cost of 19,987 ft. boards at $30.05 per M.? 
7. How much will be paid for 4 bbl. sugar, each containing 
299 lb., @ 5,¢ per lb.? 
8. 18.0327 + 4.5026. 
9. 8348 + 328. 
10. 74 A. 155 sq. rd. land at $79 per A.? 


891. Slate Exercises. 


REVIEW. | 899 


SHORT METHODS. 


7,854 x 3 9,365 x t 
1,9631 Deduct 1. 1,170% Deduct 4, 
5,8901 Ans. 8,1943 Ans. 


Multiply 6,578 by 92. 
65,780 = 10 times number. 


2,192 


2 = 4 number (Deduct). 


63,5874 Ans. 


892. Find products: 


1 
2 
3 
a 
5. 
6 
7 
8 
9 


176 x 13 11. 4,844 x 94 
273 x 12 © 12. 8,960 x 8% 
. 4,554 x $ 13. 3,245 x 7Z 
. 1,001 x 49 14. 9,060 x 114 
3,248 34 15. 658 x 994 
. 6,776 x & 16. 658 x 993 
. 2,307 x $ 17. 725 x 1194 
. 7,284 x Z 18. 347 x 798 
. 5,631 x 7% 19. 418 x 891 
9,657 x 44 20. 543 x 492 
. Multiply: 
- 418 x 99 26. 724 x 86 x 45 
. 674 x 874 27. 484 x 1° x 93 
36°. 999° 25 28. 576 x 914 x 124 
48 x 125 x 71 29. 95 x 36 x 19% 


64 x 77 x 33h 30. 74x 31x13 x 9% 


400 ARITHMETIC. 
895. Divide. Do not write products (Arts. 385, 616): 
1. 611,463,874 + 87,659 6. 703,205,104 + 71,685 
2. 279,864,597 + 45,678 7. 928,812,701 + 18,789 
8. 387,250,005 + 34,567 8. 575,646,828 + 59,764 
4. 800,700,900 = 68,439 9. 1,234,567,890 + 169,375 
5. 453,211,687 + 576,258 10. 3,126,045,000 + 483,729 


896. Write answers (Art. 385): 


11, 876,459 14, £12,000 17, 208,040 

94.317 69,999 17,613 

1p, 768,154 15, 458,124 1g, 208,018 

82.915 123,456 126,748 

654,817 375 005 862,304 

73,295 16. 59 687 87,925 
MEASUREMENTS. 


897. Slate Problems. 


1. How many boards 16 ft. long, 8 in. wide, will be required 
for a tight fence 8 ft. high, around a piece of ground 240 ft. long, 
180 ft. wide? How many posts, 6 ft. apart, will be needed? 


2. A room is 18 ft. long, 15 ft. wide. The walls and the ceil- 
ing contain 930 sq. ft. What is the height of the room? 


3. What will it cost to cover a table 6 ft. long, 24 ft. wide, 
with baize 3 yd. wide, at 75% per yd.? 


=n es 


or - 


MEASUREMENTS. 401 


4. 160 square rods make lacre. How many square yards 
are there in an acre? About how many yards long is a square 
field containing 1 acre? 


5. A 40-acre field is 160 rods long. How many rods of 
fence are needed to enclose the field? 


6. A room 80 ft. long, 24 ft. wide, 15 ft. high, contains 40 
persons. How many square feet of floor space are there for each 
occupant? How many cubic feet of air space are there for each? 


7. How many yards of carpet, 27 inches wide, would be 
needed for the floor of such a room? 


8. How many bundles of laths, each bundle covering 3 square 
yards, would be needed for the walls and ceiling of the above 
room, no allowance being made for doors and windows? 


9. A farmer owned a rectangular piece of ground 380 rods 
long, 27 rods wide. He sold three lots 18 x 8 rods, 12 x 3 rods, 
15 x 9 rods. 

Find the number of square rods in the original piece. Mark 
in the diagram the area of each lot sold, and the area of the part 
remaining. 


10. How many rods of fence will be needed to enclose the 
part remaining? 


402 ARITHMETIC. 


398. 1 gallon = 231 cubic inches. 
1 bushel = 2,150.4 cubic inches. 
lcord =128 cubic feet. 


11. How many gallons will fill a tank 22 feet long, 14 feet 
wide, 9 feet deep? 


Indicate operations. Cancel where possible. 


12. How many cords of wood are there in a pile 48 feet long, 
16 feet wide, 12 feet high ? 


BANK DISCOUNT. 


900. Wm. Brown and Sons receive the following note in set- 
tlement of their account with Thomas Tierney : 
St. Paun, May 30, 1893. 


Thirty days after date, I promise to pay to the order of Wm. 
Brown and Sons, Three Hundred Fifty-Four 743, Dollars, value 
received, at the Park National Bank. 

$ 3045455. THomas TIERNEY. 


901. This money is payable 33 days after May 30, 3 days of grace being 
allowed. If Wm. Brown and Sons desire to use the money at once, they 
may have the note discounted at a bank. In this case, the bank deducts 
from the face of the note ($354.75), the interest thereon for 33 days, and 
pays over the difference (proceeds). 

Face of note $354.75 
Discount (Int. for 33 da.) 1.95 (at 6%) 
Proceeds $352.80 


902. Slate Exercises. 


Find the discount at 6% on the following, allowing 3 days of 


grace in each case. (See Appendix, section 1305.) 
A 30-days note for $75. 


15-days note for $183.60. 
60-days note for $275.40. 
20-days note for $96. 
4-months note for $336. 


co Pw Ne 


BANK DISCOUNT. 403 


Find the proceeds, at 7%, on 
6. A 6-months note for $180. 
7. A 3-months note for $36.90. 
8. A 24-days note for $795.60. 
9. A 90-days note for $180. 
10. A 72-days note for $1,000. 


903. In computing the discount on a note, the banks ascertain the exact 
number of days. 

A 3-months note, dated February 1, is payable three days after May 1, 
which is May 4. The discount is taken for 27 +31+30+4=92days. <A 90- 
days note of the same date is payable 93 days after February 1, which is 
May 5. 

The year, however, is considered to contain 360 days; the interest in the 
first case being taken for .%%, of a year, and in the latter for 93, of a year. 


Find the discount, at 6%, on 

11. A 1-month note for $600, dated Feb. 6, 1894. (34). 
12. A 2-months note for $240, dated July 17, 1894. (485. 
13. A 3-months note for $360, dated April 8, 1894. 

14. A 4-months note for $84, dated Dec. 24, 1894. 

15. A 6-months note for $172.60, dated March 4, 1895. 
16. A 60-days note for $ 240, dated July 17, 1894. 

17. A 90-days note for $360, dated April 8, 1894. 


904. In each of the preceding examples, it has been assumed that the 
note has been presented for discount the day on which it was made. 

In some of the following examples, the notes are discounted at a later 
date, and the term of discount is to be ascertained ; that is, the time between 
the date of discount and that of maturity (including the days of grace). 

The term of discount of a 30-days note dated May 1, and discounted 
May 19, is 15 days. 


404 ARITHMETIC. 


905. In the following examples, find (a) date of maturity; (0) 
term of discount; (¢) discount; (d) proceeds: 


Dated. Face. Time. Discounted. Rate. 


18. June 16, 1894; $87.60; 30 days; July 1,1894; 6% 
19. Sept. 9, 1894; $124.18; 4months; Nov. 18, 1894; 8%. 
20. Dec. 5, 1894; $504.60; 30 days; Dec. 12, 1894; 7%. 
21. Nov. 14, 1894; $72.36; 3 months; Dec. 20, 1894; 6%. 


22. Oct. 30, 1894; $234; 90 days; Jan. dD, 18955 69%. 
23. Jan. 2, 1895; $95.90; 2months; Feb. 138, 1895; 6%. 
24. Aug. 5, 1895; $164; 60 days; Aug. 31, 1895; 8%. 


25. Feb. 27, 1895; $83.20; 100days; March 9, 1895; 6%. 


INTEREST. 
906. Find interest: 


When the time is less than one year, ascertain the exact number of days. 
When greater than a year, find the time by compound subtraction, taking 
the month of 30 days. 


1. $160; Jan. 2, 1893, to. May 16, 1896; 42%. 


3 yr. 4 mo. 14 da. = 3434 yr. = 3585 yr. = $97 yr. 


2. $342.18: April 5, 1895, to Sept. 80, 1895; 6%. 


Time, 178 days = 448 year. 


3. $59.80; Feb. 24, 1896, to Dec. 24, 1896; 5%. 
(Leap year.) 


$ 1,234.56; Aug. 3, 1890, to Jan. 1, 1896; 52%. 
$387.90; March 15, 1894, to Sept. 1, 1894; 6%. 
$96; July 6, 1894, to Feb. 4, 1895; 339%. 

$240.72: May 20, 1898, to Jan. 15, 1894; 79%. 


$983.25; Dec. 15, 1899, to March 8, 1900; 6%. 
How many days in February, 1900? Be sure you are right. 


——————— ee 


REVIEW. 405 


SHORT METHODS. 


908. When multiplying mixed numbers, many accountants prefer not 
to reduce them to improper fractions. 


909. Multiply without reducing: 


1. 433 x 84 4. 247 x BL 
300 5. 422 x 124 
Ay 6. 3535 x 84 
16 5 
2. 574 x 94 7. 562 x 44 
516 g. 838 x 34 
ity 9. 721 x 102 
ier 6 
3. 173x Th 10. 952 x 114 
910. 7634 x 92 644 x 114 
765 product by 10 7092 product by 11 
12% lessproductby 4 93, product by + 
7521 Ans. 98 3612 product by 4 
1504 Ans. 1144 
843 x 52 


421% product by 5 
16% product by 4 


16% product by 4 
4553 Ans. 52 
911. Multiply: 
11. 193 x 42 15. 514 x 6% 
4 1 1 
See iky, 16. 63} x 108 
12. 241 x 32 
(Aes) 17. 292 x 2% 
13. 352 x 73 18. 314 x 128 


(7+7 +4) 


14. 40% x 83 
(9 — 4) 20. 563 x 72 


19. 423 x 95% 


406 ARITHMETIC, 


REVIEW. 
915. Slate Problems. 
1 pound troy = 5,760 grains. 
1 pound apothecaries’ = 5,760 grains. 
1 pound avoirdupois = 7,000 grains. 


How many grains in a troy ounce? In an avoirdupois ounce? 


1. Find the value of a dozen silver spoons, each weighing 
8 oz. 5 pwt., at $1.20 per oz. (See table, Art. 828.) 


2. A gold chain weighs 384 grains. What is its cost at $1.15 
per pwt.? 


3. Add 4 lb. 6 oz. 18 gr., 5 oz. 9 pwt., 3 lb. 20 gr., and 9 lb. 
11 oz. 15 pwt. 5 gr. 


4. Gold coin contains 90 per cent gold,9 per cent silver, 1 
per cent copper. Find the quantity of each metal in 50 double- 
eagles ($20), each containing 516 grains. 


5. How many spoons, each weighing 2 oz. 18 pwt., can be 
made from 5 lb. 9 oz. 12 pwt. silver? 


6. How much money in silver dollars, 4124 grains each, will 
weigh 165 lb. avoirdupois (7,000 grains) ? 


7. What fraction of a pound avoirdupois is a pound troy? 
What per cent of an ounce avoirdupois is a troy ounce? 


8. What is the value, at $1.60 per oz. troy, of a silver pitcher 
weighing 4 lb. 8 oz. avoirdupois? 


9. At 75¥ per ounce, what is the value of the silver con- 
tained in a half-dollar, which weighs 192.9 grains, =% being pure 
silver? | 


10. What is the capacity, in gallons (231 cu. in.), of a barrel 
that will contain 24 bu. (2,150.4 ca. in.)? 


11. A tank 16 ft. long, 14 ft. wide, 8 ft. deep, is lined with 
lead on the bottom and sides. How many square feet of sheet 
lead will be required? (Draw the ‘“‘development”’ of the tank.) 


BANK DISCOUNT. 407 


12. A man sold 18 bbl. sugar, each containing 306 lb.; 21 
bbl., each containing 297 lb.; 5 bbl., each containing 291 Ib. 
What is the average weight per barrel ? 

13. A, B, and C buy a farm. A pays $8,750, B pays $ 7,200, 
C pays $4,100. What per cent of the purchase money does each 
furnish ? 


14. If 11 weavers in 9 days weave 1,584 yards, what will 1 
man doin 1 day? 6 men in 7 days? 

15. The tax rate of a certain city is 13% upon the assessed 
value of property. If this value is 75% of the actual value, how 
much taxes does Mr. Smith pay upon a house and lot, the actual 
value of which is $ 24,000? 


16. What per cent of a lb. avoirdupois is a troy pound? 


DISCOUNT OF INTEREST-BEARING NOTES. 


919. Slate Problems. 
Brooktyn, N.Y., Oct. 15, 1894. 
Sixty days after date I promise to pay to the order of Harman 
P. Payne, Forty-eight =3% Dollars, value received, with interest 
at 6%. 
$4850. GroRGE P. Post. 


1. Find the amount due on the above note at maturity. 


If the above note is discounted at a bank, the discount 1s taken on the 
amount due at maturity. 


Find the proceeds of the above note if discounted Dec. 1, 1894, 
at 6%. 
2. Find the proceeds of a 90-days note for $175, bearing 
interest at 6%, discounted 33 days after date, at 6%. 


3. Find ‘the proceeds of a 60-days note for $350, bearing 
interest at 6%, discounted at 6%, 10 days after date. 


408 ARITHMETIC. 


4. Find the proceeds of a three-months note for $840, bear- 
ing interest at 7%, discounted at bank 47 days before maturity, 
at 8%. 


5. A four-months note for $720, dated March 17, 1894, bear- 


ing interest at 6%, is discounted at 7%, May 10. What are the 
proceeds ? 


REVIEW. 
920. Slate Problems. 
1. Change 7,643 inches to rods, ete. 
2. Change 1,875 feet to rods, ete. 
3. Change 964 yards to rods, etc. 


921. English Money, 
12 pence (d.) 1 shilling (s.) 
20 shillings 1 pound (£) 
A farthing is one-fourth of a penny, and is generally written as a fraction. 
4. Reduce 4,000d. to &, s., d. 
5. How many pence in £87 17s. 6d.? : 
6. What will be the cost of 150 yd. silk at 3/6 per yd.? 
3/6 = 3s. 6d., read three and sixpence. 


7%. If £1—$4.8665, what will be the cost in U.S. money 
of 75 books at 18 pence each? 


8. A merchant sells 37 coats at £3 5s. each, less 10%. 
What is the amount of his bill in English money? 


9. Find 25% of £183 14s. 8d. 


10. A silver dollar weighs 412% grains. How many ounces 
of pure silver are there in 1,000 silver dollars if the coin is 
pure silver? 


11. The one-cent pieces weigh 48 grains. How many dollars 
would weigh 120 pounds avoirdupois (7,000 grains to pound)? 


REVIEW. | AQ9 


12. A coal dealer buys 150 tons of coal, 2,240 pounds each, at 
$4.50 per ton. He sells it at $4.75 per ton, giving 2,000 lb. to 
the ton. What is his profit? 


13. Three workmen receive $283.50 for doing a piece of work. 
One worked 32 days, the second worked 53 days, the third 
worked 41 days. What is the share of each? 


14. Three men engage in a business venture. One furnishes 
$3,000, another furnishes $5,000, a third furnishes $4,000. They 
gain $1,800. What is each one’s share of the profit? 


15. A garrison of 1,200 men has rations for 40 days. Ten 
days later it receives a reinforcement of 8300 men. How many 
more days will the rations last? 


16. Two trains start at 9 o’clock from towns A and B, 120 
miles apart. The train leaving A travels 20 miles per hour, the 
other 30 miles per hour. What time do they meet, and at what 
distance from A? 


17. Divide $900 among four persons, so that the second will 
have three times as much as the first; the third, twice as much 
as the second; the fourth, as much as the three others. 


18. Divide $540 among three persons, so that the first will 
have $48 more than the second, and the second $75 more than 
the third. 


19. Four men working 7 hours a day need 15 days (105 
liours) for a piece of work. How many days would it take 6 
men, working 10 hours per day, to do the same work? 

20. A grocer buys some eggs at 15¢ per dozen. He breaks 
15, and then finds that by selling the rest at 16% per dozen, he 
will neither gain nor lose. How many eggs did he buy? 

21. If a person lends me $250 for 8 months, for how long 
ought I to lend him $400 as an equivalent? 

22. A fort had a garrison of 4,000 men, and provisions for 18 
weeks. If 1,000 men were sent away, how long would the pro- 
visions last? 


410 ARITHMETIC. 


COMMERCIAL DISCOUNT. 
924. Slate Problems. 


1. On a bill of goods amounting to $583.40, a discount of 
5% is given for cash. What is the amount paid? 


2. What will be the cost of 16 gross of Roman candles at 
$26.75 per gross, less 60% ? 


8. Sept. 1, 1894, 1 bought tea amounting to $1,876.50. If 
5% is deducted for payment within ten days, how much would I 
have to pay if I paid the bill Sept. 9? 

4, What will be the cost of 15 cases cocoa @ $18.20 each, 
less 20% ? 

5. Bought 5 gross of essence of lemon at 50% per doz., less 
5%. What is the amount of my bill? 

6. Find the cost of 15 cases of chloride of lime, 50 lb. per 
case, at 91 ¢ per lb., less 15%. 

7. What will be the net cost of a bill of plated ware amount- 
ing to $84.75, on which a discount of 334 and 10% is allowed? 


$84.75 
less 4 28.25 
56 50 This means 331% discount on the gross amount. 
{ and 10% discount on the remainder. 
less =1, 5.65 
Ans. $ net. 


8. Find the difference between $390 less 431% discount, 
and $390 less 3834 and 10% discount. 


9. Thurber, Whyland Co. sold the following goods. Make 
out the bill, less 50 and. 10 and 10 and 10% discount. 


500 1-pound bags $1.00 per he 


1,500 nee i fee 
8,000 1spounds ew 60? Tan 
5500 ‘Te ponnid ae sere Ost eee 


2,000 S2pound yes EZ U0 es 


MEASUREMENTS. 411 


10. Find the net cost of 18,500 bags at $4.40 per M. less 60 
arid 10 and 5%. 


11. Which is the better discount for the buyer, 40 and 10% 
or 80 and 20%? What will be the difference on a bill of $100? 


12. $100 less 834 and 10% discount is equal to what? What 
per cent discount is 334 and 10% equal to? To what per cent 
net is it equivalent? 

926. Oral Problems, 

1. A piano, marked $800, is sold at a discount of 25 and 
10%. What is the selling price? ) | 

2. Bought goods amounting to $ 600, less 5% for cash. What 
is the net cost of the goods? 

3. What single discount is 50 and 10% equal to? 

4. What single discount is 80 and 30% equal to? 

5. Paid $729 for goods, on which 10% was allowed. . What 
was the “‘gross’’ price? 

6. How much will I have to pay for 12 doz. bottles flavor- 
ing extract, at 60 per doz., less 10% ? 

7. Whatis the “list” price of an article for which I paid $48, 
after a discount of 25% was deducted? 


MEASUREMENTS. 
929. Slate Problems. 


1. Mark in each division of each of the following rectangles 
its area in square feet. Dimensions of each rectangle 100 ft. by 
60 ft. 


412 ARITHMETIC. 


2. Find the area of each of the following triangles in eae 
feet; base of each 100 feet, altitude 60 feet. 


3. Mark in each of the following eleven triangles its area in 
square feet; altitude of each, 60 ft. 


4. Find the area of each of the following four triangles in 
square feet; altitude of each, 60 ft. 


5. Find the area of each of the following four Dara Reba 
altitude of each, 60 ft. 


40 | { 
: Nas 
S| Nas 

! 1 40 


6. Find the area of each of the following trapezoids; altitude 
of each, 60 ft. The number of square feet in each is equal to 60 
ie by what? 


ae 
! as ! ! 
I bead Seine |7, 
as : ont in 
20 | 1 30 1) 


ceil il 


SURFACES AND VOLUMES. 413 


7. One diagonal of each of the following quadrilaterals meas- 
ures 100 feet. The perpendiculars let fall on this diagonal from 
the opposite corners measure 30 ft. and 40 ft., respectively. Find 
the area of each in square feet. 


8. Cut from a strip of paper, AB, two inches wide, a rectangle, 
a rhombus, and a rhomboid, as given in the accompanying dia- 
gram. Show that the three parallelograms are equal in surface. 


3 in, 3 in, 


ow aa 


3in. ne. 


9. Cut from a strip of paper, two inches wide, three trapezoids. 
Make one parallel side of each 2 inches long, and the other 
parallel side 3 inches long. Divide up each trapezoid in such 
a way as to show that its surface is equal to that of a rectangle 
24 in. by 2 in. 


Qin. Qin, Qin, 24 in. 
3 in. 3 in, Zin, 24 in, 


SURFACES AND VOLUMES. 
930. Slate Problems. 


1. How many cubic feet are there in a block of granite whose 
base is 44 feet square, and whose height is 6 feet ? 


2. Find the value of the above block at 60¢ per cubic foot, 
and the cost of polishing its six faces at 60¢ per square foot. 


414 ARITHMETIC. 


3. A contractor charged 80% per cubic yard for digging a 
cellar 21 feet wide, 60 feet long, 9 feet deep. What was the 
amount of his bill? 


4. What will be the cost of cementing the floor of the above 
cellar at 75% per square yard? Deduct from the given dimen- 
sions the thickness of the four walls, eighteen inches each. Make 
a diagram. 

5. Acubic foot of water weighs 1,000 ounces. Marble is 2.8 
times as heavy. Calculate the weight, in tons, of a marble shaft 
4 feet square and 12 feet high. (Cancel.) 


6. A carpenter is making a cubical box whose inside meas- 
urement is 1 foot. Each side consists of a single piece of wood 1 
inch thick. Give the dimensions, in inches, of each of the six 
pieces used. 


7. How many pounds avoirdupois would there be in a brick 
of pure gold 8 x 4 x 2 inches, gold being 19.4 times as heavy as 
water ? 


8. Find the weight in pounds of a cord of pine wood, which 
is .66 times as heavy as water. 


9. A cube of marble, 12 inches on a side, is enclosed for 
transportation in a tightly fitting wooden box made of material 
one inch thick. What are the outside dimensions of the box? 
How many cubic inches in the wood and the marble together? 
How many cubic inches are there of each? 


10. An iron cube 2 feet long weighs 2 tons. How many times 
as heavy as water is iron? Calculate the weight of an iron cube 
1 foot long. Of one 3 feet long. 


CHAPTER XII. 


SIMPLE AND COMPOUND INTEREST. — DISCOUNT. — CAUSE 
AND EFFECT. — PARTNERSHIP.— BONDS AND STOCKS. 
— EXCHANGE.— LONGITUDE AND TIME. — SURFACES 
AND VOLUMES, 


TO FIND PRINCIPAL, RATE, OR TIME. 


931. At what rate per cent will $723.60 amount to $759.78 
in 1 yr. 1 mo. 10 da.? 


Let 2 = rate. 
Then 723.60 x A x 429 — 8.042 = interest. 
723.60 + 8.042 = amount = 759.78 
Transposing, 8.04% = 759.78 — 723.60 = 36.18 
Clearing of decimals, 804a = 3,618 
v= 3618 — 41 Ans. 41 per cent. 


932. In what time will $85.50 produce $8.17 interest, at 4 
per cent ? 


Let x = time in years. 

Then 85.50 X 745 X @ = 3.42 x = interest. 
3.42 2 = 8.17 

Clearing of decimals, 342 % = 817 


ao = §17—= time in years. 
2 yr. 4 mo. 20 da. 
342)817 yr. 
133 yr. remainder. 
12 
1596 mo. new dividend. 
228 mo. remainder. 


30 
6840 da. new dividend. 
0) Ans. 2 yr. 4 mo. 20 da. 


415 


416 ARITHMETIC, 


933. Slate Exercises, 
Iind rate, time, etc. 
Principal, $2,000; time, 3 yr.; interest, $300. Rate? 
Principal, $1,800; rate, 4% ; interest, $144. Time? 
Time, 8 mo.; rate, 44%; interest, $2.88. Principal ? 
Principal, $38; time, 2 yr.; amount, $40.28. Rate? 
Principal, $140; rate, 34%; time, 3 mo. 15 da. Interest? 
Amount, $39.60; rate,4%; time, 2 yr.6 mo. Principal? 
Amount, $484.15; rate, 834%; principal, $460. Time? 
8. Principal, $39.60; rate,4%; time, 1 yr. 7 mo. 15 da. 
Amount? 
9. Time, 8 yr.; rate, 3%; amount, $6,200. Principal? 
10. Principal, $7,548; time, 3 mo. 5 da.; interest, $119.51. 
Rate ? 
11. Principal, $9,000; rate, 4% ; interest, $632. Time? 
12. Time, 2yr. 3 mo. 20 da.; rate, 5%; amount, $160.60. 
Principal ? 
13. Principal, $756; rate, 31%; time, 3 yr. 4 mo. 20. da. 
Interest ? 


14. Principal, $120; time, 1 yr. 2 mo. 15 da.; interest, $4.35. 
Rate ? 


15. Amount, $97.57; rate, 4%; interest, $7.57. Time? 


16. Time, 3 yr. 8 mo. 19 da; rate, 44%; amount, $93.39. 
Principal ? 

17. Principal, $1,848; rate, 38%; time, 4 yr. 9 mo. 25 da. 
Amount ? 


18. Rate,5%; time, 4 yr. 6 mo. 28 da.; interest, $16.43 
Principal ? 


AEs pape ht he he 


INTEREST. 417 


934. Oral Exercises. 
In what time will $100 amount to $109, at 6% interest? 
At what rate will $200 produce $16 interest in 2 years? 


What principal will produce $ 12 interest in 8 years, at 4%? 
In what time will $300, at 4%, produce $29 interest ? 

In what time will $170 produce $1.70 interest, at 5% ? 
In what time will $360 produce $3.60 interest, at 4%? 
In what time will $725 produce $7.25 interest, at 6% ? 
In what time will $45 produce 45 interest, at 44% ? 

In what time will $72 produce $1.44 interest, at 6%? 
Find the interest on $84 for 144 days, at 5%. 

. Find the interest on $125, at 5%, for 2 months 12 days. 
. At what rate will $64 produce 64¢ interest in 80 days? 
13. At what rate will $40 produce $1.20 interest in 6 months? 


14. A certain principal produces $120 interest, at 6%. What 
would be the interest if the rate were 4%? 


ononnrt on FF WO WD 


—-— = 
ue oe) 


INTEREST BY ALIQUOT PARTS. 


936. Slate Exercises. 


1. Find the interest on $387.45, for 2 yr. 8 mo. 18 da., at 
7%. 
i $387.45 x .07. 
$ 27.1215 interest for 1 year. 
27.1215 interest for 1 year. 


6 mo. = 3 yr. 13.5607 interest for 6 months. 
2mo.=i(of 6mo.) 4.5202 interest for 2 months. 
15 da. =4(of 2mo.) 1.1301 interest for 15 days. 
1 


3 da. = ¢ (of 15 da.) .2260 interest for 3 days. 
Ans. $73.68 interest for 2 yr. 8 mo. 18 da. 


418 ARITHMETIC. 


2. Find the interest on $432.90, at 6%, for 1 yr. 7 mo. 12 da. 


$432.90 x .06. 
interest for 1 yr. 


6 mo. = 4 yr. interest for 6 mo. 
1 mo. =i (of 6 mo.) interest for 1 mo. 
10 da. = 4 (of 1 mo.) interest for 10 da. 
2 da. = + (of 10 da.) interest for 2 da. 


interest for 1 yr. 7 mo. 12 da. 


3. Find the amount of $874.16, at 5%, for 1 yr. 9 mo. 4 da. 
$874.16 principal. 


5% = dy 43.708 interest for 1 yr. 
6 mo. = 3 yr. interest for 6 mo. 
3 mo. =4 (of 6 mo.) interest for 3 mo. 
3 da. = z5 (of 3 mo.) interest for 3 da. 
1 da, =2 (of 3 da.) interest for 1 da. 


amount for 1 yr. 9 mo. 4 da. 


4. What is the amount of $95.72, for 3 yr. 6 mo. 20 da., at 
5%? 


$95.72 principal. 


(10% =) x5 9.572 interest for 2 yr. 
lyr. = } (of 2 yr.) 4.786 interest for 1 yr. 
6 mo. = 4 yr. interest for 6 mo. 
20 da. = ? of 6 mo. interest for 20 da. 


amount for 3 yr. 6 mo. 20 da. 


5. Interest of $1,806.45, at 4%, for 1 yr. 7 mo. 25 da. 
1 yr., 6 mo., 1 mo., 15 da., 5 da., 5 da. 


6. Interest for 10 mo. 29 da., at 4%, on $380.40. 


$380.40 x .04. 
$ 15.2160 interest for 1 year. 
1 mo. = + yr. interest for 1 month ) deduct from interest 
1lda. = J; mo. _______ interest for 1 day for 1 year. 


interest for 10 mo. 29 da. 


INTEREST. 419° 


7. Amount, at 6%, of $125.73, for 2 yr. 10 mo. 4 da. 

8. Interest on $84.66, at 7%, for 1 yr. 4 mo. 12 da. 

9. Interest, at 5%, for 4 yr. 2 mo. 7 da., on $250. 
10. Amount of $1,000, at 6%, for 338 days. 


937. When the time is less than a year, the following facts 
should be remembered: 


6% for a year is 1 per cent for 60 days. 
59, for a year is 1 per cent for 72 days. 
41% for a year is 1 per cent for ? days. 
4% for ayearis1 percent for ? days. 


11. Find the interest for 81 days, at 5%, on $876.40. 


72 days = 1% = $8.764 
9 days = }-(of 72 da.) = 1.095 
~9.86 interest for 81 days. 


12. Amount of $954, at 4%, for 4 mo. 10 da. 


Principal $ 954. 
3 months’ interest =1% 9.54 
1 mo. = 4 (of 3 mo.) 3.18 
10 da. = 4 (of 1 mo.) 


amount for 4 mo. 10 da 


13. Interest of $1,874, at 44%, for 935 days. 


80 days = 1%. 
10 days 

2 days 

1 day 


14. Interest of $753.20, at 5%, for 158 days. 


72 da., 72 da., 12 da., 2 da. 


15. Amount of $1,234.50, for 193 days, at 6%. 
60 da., 120 da., 12 da., 1 da. 


420 ARITHMETIC. 


16. Find the proceeds of a 90-days note, for $873.60, at 6%. 


Face $873.60 
60 da. 8.736 
30 da. 4.368 Deduct. 
3 da. A37 
$860.06 proceeds. 


17. Find the discount on a 38-months note, for $1,596, at 6%. 


18. What are the proceeds of a 6-months note, for $785, 
discounted at 6%. 


19. Find the interest on $484.40, for 1 yr. 3 mo. 17 dal at 7%. 
20. Find the amount of $683, for 3 yr. 4 mo. 11 da., at 44%. 


938. N.B.—Do not use unnecessary figures. 

21. Principal, $360; 5%; 3 yr. 7 mo. 18 da. Interest? 
22. Principal, $613; 44%; 157 da. Amount? 

23. Principal, $1,774; 832%; 17 mo. 23 da. Interest? 
24. Principal, $875; 6%; 2yr.3mo.1da. Amount? 
25. Principal, $976; 7%; 325 da. Interest? 


939. By the time of a note is meant the number of days, etc., for which 
it is drawn. Find the discount for three additional days. 


26. Face of note, $254; time, 80 days; 7%. Proceeds? 
27. Face of note, $515; time, 6 months; 5%. Discount? 
28. Face of note, $493; time, 60 days; 8%. Proceeds? 
29. Face of note, $717; time, 15 days; 64%. Discount? 
30. Face of note, $1,000; time, 90 days; 4%. Proceeds? 


940. Find the exact number of days. Take 360 days to year. 
31. Principal, $1,886.50; 6%; Jan. 2 to Dec. 1. Amount? 
32. Principal, $1,295.70; 7%; March 8 to April 9. Interest? 


COMMERCIAL DISCOUNT. 491 


33. Principal, $1,433.11; 5%; Feb. 13 to Sept. 4. Amount? 
34. Principal, $765.90; 4%; Oct. 1 to Dec. 17. Interest? 
35. Principal, $275.84; 51%; May 9 to July 3. Amount? 


941. By the term of a note is meant the number of days it has to run 
after it has been discounted, including days of grace. 


36. Face of note, $100; term, 63 days; 7%. Discount? 
37. Face of note, $200; term, 93 days; 64%. Proceeds? 
38. Face of note, $300; term, 24 days; 54%. Discount? 
39. Face of note, $400; term, 117 days; 8%. Proceeds? 
40. Face of note, $500; term, 88 days; 5%. Discount? 


942. In examples 41-45, inclusive, find the time by compound sub- 
traction. 


41. Principal, $25.83; 6%; Jan. 14, 1892, to Sept. 5, 1894. 
Interest? 


42. Principal, $47.96; 5%; Feb. 6, 1898, to Aug. 1, 1896. 
Amount? 

43. Principal, $85.30; 7%; March 25, 1894, to Jan. 18, 1897. 
Interest? 

44. Principal, $75.00; 4%; April 15, 1888, to Feb. 6, 1895. 
Amount? 


COMMERCIAL DISCOUNT. 


944. Slate Problems. 
1. Ona bill of goods amounting to $ 874.40, a discount of 5% 
is allowed. How much must be paid? 


$ 874.40 
5%= gy 43.72 
$ 830.68 Ans. 


Divide by 2, placing the quotient figure one place to the right ef the cor- 
responding figure of the dividend. 


429 | | ARITHMETIC. 


2. Find the cost of a wagon, the catalogue price of which is 
$750, the discount being 30%. 


$750 x .70 = $525. Ans. 
The net cost'is .70 of the catalogue price. 


3. What will be the cost of goods amounting to $1,837.60, 
on which there is allowed a discount of 174%? 


$ 1,837.60 
10% = 74 183.76 
5%=1 (of 10%) 91.88 Deduct. 
214 =1(of 5%) 45.94 
$1,516.02 Ans. 


For 10% rewrite the original amount, placing first figure one place to the 
right. 5% is 4 of 10%. 23% is 4 of 5%. 


4. $784.68 less 75%. 
1 of $784.68 = Ans. 
5. $937.52 less 36%. 


$937.52 
25% = 4 234.38 
Deivet} 10% = #5 of $937.52 
1G = 


6. Find the net cost of 1,630 yd. silk, invoiced at $1.10 per 
yard, less 16% discount. 


7. What is the cost, in francs, of 848.72 meters silk, at 5.75 
francs per meter, less 12% ? 


8. What is the net cost of a lot of musical instruments 
amounting to $1,875.60, on which a discount of 10, 5, and 24% 
is allowed? (Art. 924, Ex. 7.) 


9. What would be the net cost of the same articles, if the 
discount were 24, 5, and 10%? 


10. Find the net cost of the same, at 174% discount. 


11. Goods catalogued at x dollars are sold at a discount of 20 
and 10%. Find value of z, if net price is $ 360. 


REVIEW. 493 


946. Oral Problems. 


1. A can do a piece of work in 5 hours, Bin 7 hours. How 
long will it take both working together? 


2. An agent collected a bill, and sent to his employer the 
amount, less 24% commission. If his commission was $1.60, 
how much did he remit to his employer? 


3. My house, worth $12,000, is insured for 2 of its value, at 
1%. What premium do I pay? 

4. A floor 6 yards long, 4 yards wide, needs 32 yards of 
carpet to cover it. What is the width of the carpet? 

5. What will be the interest on $87, at 5 per cent, for 144 
days? 

6. Find the discount, at 8 per cent, on a note for $176, 
which has 90 days to run. 


7. An agent receives $8,200 to invest after deducting his 
commission of 75 of the amount invested. What is the agent’s 
commission? 


8. By selling a house for $3,500, I lose $500. What is my 
loss per cent? 


9. A lot is sold for $1,200, at a loss of 20 per cent. What 
part of $1,200 is the loss? 


10. A merchant’s receipts are $1,200; his gain is 20 per cent. — 
What part of his receipts is profit? 


11. If 3 men earn $72 in 8 days, how many dollars will 5 
men earn in 11 days? 


12. Ifa dealer loses 25% by selling a horse for $225, what 
per cent would he gain or lose by selling the horse for $325? 


13. Find the cost of 4 yd.1 ft. of ribbon, when 2 yd. 2 ft. 
cost 40 cents. 


424 ARITHMETIC. 


TO FIND FACE OF NOTE, RATE OF DISCOUNT, AND TIME. 


947. I wish to obtain $1,000 from a bank. What must be 
the face of a 30-days note, which will give the above proceeds, if 
it is discounted at 6% ? 

Let x = face of the note. 


6 33 Ad we discount. 


100°. 360 2000 


ae te proceeds = 1,000 
2000 


Clearing of fractions, 2,000 a — 11 « = 2,000,000 
1,989 2 = 2,000,000 


_ 2000000 
1989 


Ans. $1,005.53, face of note. 


= 1,005.53 


Proof. Face of note, $1,005.53 
30 days’ discount = 5.0276 ) 4%. 
Deduct { 3 days’ discount = 5027 75 of 30 days. 


Proceeds $ 1,000.00 


948. A note for $1,980 was discounted at 6%. The proceeds 
ryere $1,959.21. How many days had the note to run? 


Let x = term in days. 
1,980 x su x eo $3 discount. 
100 360 J%0 
1,980 — 23 — proceeds = 1,959.21 
100 
Clearing of fractions, 198,000 — 33a” = 195,921 
— 33a = — 2,079 
x = 63 


Ans. 63 days. 


BANK DISCOUNT. TADS 


949. Slate Exercises, 


1. Three-months note; face, $108; rate 6%. Find pro- 


ceeds. 
(Term of discount is 93 days.) 


2. 90-days note; face, $360; discount, $6.51. Find rate. 

3. Proceeds, $717.60; rate,5%; face, $720. Find term. 

4. Discount, $11.20; rate, 7%; term, 48 days. Find face. 

5. 15-days note; face, $1,560; rate, 6%. Find discount. 

6. Term, 20 days; face, $158.40; proceeds, $157.96. Find 
rate. 

7. Rate, 7%; discount, $2.10; face, $150. Find term. 


8. Two-months note; discount, $14.70; rate, 7%. Find 
face. 


9. For what amount must a 60-days note be drawn, so that 
the proceeds will be $300 when the rate of discount is 8 per 
cent ? 


10. A note for $120 was discounted at a bank March 15, 
1894. What is the date of the maturity of the note, the pro- 
ceeds being $119.52 and the rate of discount 6 per cent? 


11. Find the proceeds of a 6-months note for $875 drawn 
Jan. 2, 1894, and discounted at 6 per cent 35 days after that 
date. 

12. A merchant bought 300 barrels of flour at $4.75 per bobl., 
cash, and sold it for $5 per bbl., taking in payment a 60-days 
note for the amount. If he has the note discounted immediately 
at a bank, at 7 per cent, what does he gain by the transaction? 

13. What will be the face of a 30-days note (without grace), 


the proceeds of which when discounted at a bank at 6%, will 
pay for 8,000 bu. corn at 492¢ per bushel ? 


14. The proceeds of a note for $1,200, due March 15, 1896, 
and discounted at 6%, were $1,184.80. When was it discounted? 


4.26 


ARITHMETIC. 


SPECIAL DRILLS. 


951. Find sums: 


23 + 37+ 48 
44 + 66+ 19 
75 + 42 + 37 
16 + 71 + 62 


$7.56 + $5.38 
$9.74 + $8.54 
$3.49 + $9.89 
$4.83 + $6.52 


952. Find remainders: 


90—34—89 $63.20—$4850 $98.63 $75.21 
94-27 —66 $27.80—$19.90 $63.44 — $50.20 
85 —42—87 $34.10—$17.30 $86.75 — $42.50 
79--16—12 $56.70—$20.70 $73.24 —$31.10 


1,300 — 654 
1,295 — 986 
1,111 — 777 
1,463 — 684 


953. Find products: 


RD eS) 
82 x 19 
(33x 19 
64 x 19 


BU x20 
42 x 29 
32 X 29 
23 X 29 


954. Find quotients: 


378 +18 
462+ 14 
475 +19 
448 +16 


256 + 16 
289 + 17 
493 +17 
465 +15 


955. Find answers: 


136 x $ 

290 x 3% 
315 x 14 
378 xX 14 


6414 +5 
732 + 8 
| 882+ 7 
471 +9 


52+ 41 + 34 
28 + 38 + 43 
81 + 49 + 24 
63 + 47+ 33 


AES 
88 x 99 
49 x 99 
56 x 99 


468 +18 
900 + 75 
675 + 75 
975 + 75 


22, x 194 
24 x 193 
82 x 29% 
45 x 914 


325 + 865 
472 + 935 
567 + 629 
784 + 796 


67 x 101 
78 x 101 
89 x 101 
98 x 101 


175 + 124 
75 + 61 
675 + 374 
875 + 624 


654-+13 
1094 +12 
160 +14 
184-14 


REVIEW. 497 


956. Oral Problems, 


1. Paid 23¢ for calico, 27¢ for ribbon, and 48¢ for collars. 
What was the amount of my bill? 


2. A farmer had 95 sheep. He sold 39, and 17 died. How 
many had he left? 


3. What will be the cost of 16 base balls, at 49% each? 


4. How much paint will there be in 27 casks, each contain- 
ing 75 l|b.? 


5. A man divided a 429 acre farm into plots of 13 acres 
each. How many such plots were there? 


6. There are 900 men in a certain regiment. How many 
companies of 75 men each are in the regiment? 


7. Find the cost of 186 Ib. sal-soda, at Z¥ per lb. 


8. At 191¢ per yd., what will I have to pay for 64 yd. 
gingham ? 


9. How many square inches in a sheet of paper 104 inches 
long by 44 inches wide? 


10. If 22 yards of cloth are needed for a jacket, how many 
jackets can be made from 182 yd.? 


11. How many yards around a field 96 yards long, 75 yards 
wide? 


12. What will be the area, in square rods, of a triangle 33 
rods base, altitude 42 rods? 


13. How many acres in 4,960 square rods? 
14. How many feet in a mile? 


15. I paid $16.25 for cloth at $1.25 per yard. How many 
yards did I buy? 


16. What will be the cost of 8 1b. 7 oz. of tea, at 64 per lb.? 


498 ARITHMETIC. 


17. Half a number +4 of the same number = 85. What is 
the number ? 


18. I mix 4 lb. of coffee costing 20%, with 6 lb. costing 25 ¥. 
What is the mixture worth per lb. ? 


19. A tailor makes up 99 yd. cloth into trousers, using 2% yd. 
per pair. How many pairs of trousers does he make? 


20. How many feet in 21 rods? 


21. At 60 per pound, what will be the cost of a chest of tea 
weighing 45 lb.? 


22. A man owns a strip of land with a frontage of 576 feet. 
How many lots 18 feet front can he make? 


23. How much will be paid for 21 |b. butter, at 28% per lb. ? 


SHORT METHODS. 
Slate Exercises, 
6,748 
x 427 
47236 Multiply by 7. Multiply this product by 6. Why? 
2 834 16 
2,881,396 


958. Find products: 


1,193,925 X'328 6. 31,265 x 164 
2. 12,345 x 278 7. 5,763 x 426 
Ben 2.08 (C287 8. 87,093 x 486 
4. 20,508 x 142 9. 6,905 x 364 
5. 4,321 x 189 10. 64,271 x 357 
3,289 
832 ; 
36312 Multiply by 8. Multiply this product by 4. Where is 
105 248 the second product placed? Why? 


2,736,448 


d1. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


4,008 x 214 
8,736 x 742 
8,764 x 327 
1,087 x 848 
8,319 x 416 
6,352 x 927 
2,781 x 525 
9,060 x 1,166 
6,329 x 618 
2,345 x 1,272 


95.00 
734.18 
.69 

3.75 
28.14 
1,059.23 
22,965.89 
387.42 
1,369.78 
(A OSEREY 
83,008.08 
699.69 
88 

3.86 
50.05 


$ 2,000,000.02 


959. Find products: 

21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 


REVIEW. 


7,214 x 99 

8,281 x 999 
6,085 x 75 

6,984 x 25 

5,796 x 622 
8,883 x 124 
3,428 x 874 
7,154 x 874 
6,419 x 334 
6,208 x 663 


960. Supply missing amounts: 


41. $834,682.50 42. $16,945.84 


123,456.78 
9,876.54 
385.89 
57.40 

98 

7.28 

16.84 
275.30 
8,888.88 


64,935.27 
148,376.95 
834.11 
2,070.08 
12,316.99 
7,456.83 

$ 456,789.01 


31. 
32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 
40. 


43 


876 x 9F 

547 x 192 
734 X 294 
615 x 3894 


429 


AZT OMT Lise G6 
284 x 81x 19 


876 x 272 
973 x 244 


5,147 x 126.8 


4,284 x 4514 


. $380,086.77 
64,593.25 
8,737.84 
695.27 

47.16 


96.86 
408.08 
2766.59 
32,059.87 
165,384.26 
32,564.37 
6,999.88 
840.30 
27.63 

5.98 

86 
$743,869.05 


430 ARITHMETIC, 


44, £7 16s. 7d. 45. 8yd.2ft. Gin. 46. 6bu.3 pk. 5qt. 
9 yd. 1 ft. 10 in. 


4 18s. 11d. 5 bu. 2 pk. 7 qt. 
£120" 18.) Sd. 30 yd. O ft. 3 in. 25 bu. 1 pk. 1 qt. 


961. Divide. 2 decimal places (Arts. 385, 616): 
47. 46,893,647 + 3,986,048 50. 76,538,061 + 5,786,804 . 
48. 26,053,862 + 1,898,637 51. 92,647,318 + 4,863,978 
49. 38,627,000 + 2,679,835 52. 57,913,246 + 2,597,384 


962. Write answers (Art. 385): 


53, 46,898,647 _ ge, 76,588,061 _ 
9,728,759 8,736,804 

54, 26,058,862 _ wy, 92:647,818 _ 
2.198,684 9,863,978 

55, 38:627,000 _ gg, 57,918,246 _ 
3,568,879 7,384,597 


TABLE OF EXPORTS. 


963. The following table contains the values of the goods 
exported in 1891 by the United States to the various countries 
of the world. 


Find the total value of the goods exported, and the per cent of this value 
for each section. Carry out to two places of decimals. 


Countries to which Exported. 1891. Per cent. 
A arope ee 2 oe sl eae at Oe ane eee Fy bese ? 
2. Asia and Oceanica .°. . Main hice 43,813,519 ? 
3. British North American eee aay ee 37,345,515 ? 
ACW OBE LI GIGS |, sav tuiite ab cual ewe o diee nee Seana meats 33,416,178 : 
5. South America . . 33,226,401 ? 
6. Mexico, Central Ne A ns Her intAs 21,236,545 ? 
TN ERICA a i! ght en ee ca re Be 4,738,847 ? 
SA TG USL ay ies Se Wipe ae Ae ie a Be 879,172 ae 


? 100.00 


INTEREST, 431 


REVIEW. 
964. Oral Problems. 


1. 3 ofa number is 48. What is the number? 


2. A base ball club won 17 games, and lost 13 games. What 
per cent of its games did it win? ; 


3. What per cent of 4 is 64? 
4. 221s what per cent of 34? 


5. How many acres in a rectangular farm 1 mile long, 4 
mile wide? 


6. What per cent of the “list” price is paid by a buyer who 


receives a discount of 20 and 10 per cent? 


7. A tank is filled by two pipes, one of which can fill it in 
6 hours, and the other in 8. How long will it take both together 
to fill the tank? 


8. Find the interest on $80, for 72 days, at 6%. 


9. A man sold a wagon for $ 420, which was 16% less than 
it cost. How much did he lose? 


10. A kilo is 2.2046 lb. How many pounds in 1,000 kilos? 


965. Slate Problems. 


In the first four examples, carry out to two places of decimals, as: 
135.67 %, 25.83 %, 6.03%, 0.52%, 0.09 %. 


1. The population of Montana was 39,159 in 1880, 132,159 
in 1890. Find the per cent of increase. 


2. The population of South Dakota in 1890 was 328,808, a 
gain of 230,540 over the population in 1880. What was the 
gain per cent? 


3. The enrollment in the South Dakota schools was 9,972 in 
1880, 66,150 in 1890. Find the gain per cent. 


432 ARITHMETIC. 


4. What was the gain per cent in the population of a state 
that had 416,396 inhabitants in 1880, and 416,552 in 1890? 


5. A man marks an article $1.50, and sells it at a discount 
of 25% from the marked price. If the article cost him 90, 
what is his gain per cent? 


6. Goods costing $8 are sold at an advance of 20 per cent. 
The marked price is $12. What per cent reduction is made on 
the marked price? 


7. A rug costs $20. It is sold at a profit of 20%. The 
selling price is 20% below the marked price. How much is 
received for the rug? What is the marked price? 


8. What price must cloth, which costs $2 per yard, be 
marked so that a profit of 20% will be made when the cloth is 
sold at 20% less than the marked price? 

REVIEW FRACTIONS. 


969. Slate Exercises, 
NorE. — Do not use too many figures. 
1. Add 4, 24, 2, 2. 
2. Divide each of the following fractions by 6: 
HTS 3b tos 
3. Reduce 4 of 5% of 33; of 27 to a simple fraction. 
4. 383— 2111. 
5. What fraction of £1 18s. 9d. is 5s. 6d. ? 
6. Multiply 244 by ¢ of #. 
7. What is the greatest common divisor of 657 and 1,168? 
The least common multiple of 12, 16, 20, 30? 
8. What must be taken from 8,5 to leave 374? 


9. Reduce $32 and 428 to their lowest terms. 


10. Which is the greatest and which is the least, of 4, ¢ of 3, 
and 24 of =4-? 


1A 
12. 
13. 


14. 
LS; 
16. 
ic 


MEASUREMENTS. 433 


What must be added to 3,4, to make 54? 
Add 2 of a week, 2 of an hour, 7% of a minute. 
How much is 9 times each of the following fractions? 


AP By Et 
&) 11) 27) 63° 


302 +3 of 7. 

Tz t+ % of py + Z of F. 

What part of a ten-acre field is 4 A. 100 sq. rods? 

What is the least number that will contain each of the 


numbers 6, 15, 18, and 20? 


18. 


19. 


20. 


21. 


22. 


23. 
24. 
25. 


What must be multiplied by 44 to produce 161? 


What is the value of Tak 


4 
What quantity must be divided by 42 to produce 88? 


Find the value of ai. 


12 6 


How much is tot of 8 da. 15 hr. 82 min.? 


6 
Reduce ;4; mile to rods. 


Add 2, 4,52. Subtract 4,5; from the sum. 
Multiply 3 of 51 by 74. Divide the result by 14. 


MEASUREMENTS. 


970. Slate Problems. 


1. A tank 18 ft. long, 15 ft. wide, requires 63 sq. yd. of 
lead to line its sides and bottom. How many feet deep is it? 


2. 


(Make diagram.) 


A farmer has a 38-acre field in the form of a right-angled 


triangle. If one perpendicular side measures 242 yards, what is 
the length of the other? 


434 ARITHMETIC. 


3. One parallel side of a field in the shape of a trapezoid 
measures 150 yd., the other measures 200 yd. How many 
square yards in the field, the perpendicular distance between the 
sides being 50 yards? (Make diagram.) 

4. The shorter parallel side of a trapezoid is x yards, the 
other is 100 yd., the perpendicular is 60 yd. Find the area in 
square yards. 

How long is the shorter parallel side when the area is 5,400 
sq. yd.? 

5. One parallel side of a trapezoid is 80 yd., the other is 
120 yd., the perpendicular is x yd. Find the area in square 
yards. 

How long is the perpendicular when the area is 4,000 sq. yd.? 


6. One parallel side of a trapezoid is x yd., the other is 
z+ 40 yd., the perpendicular is 60 yd. Find the area. 
Find the length of the parallel sides when the area is 6,000 
sq. yd. 
7. The sum of the parallel sides of a trapezoid is 200 yards, 
the perpendicular is 100 yards. How many square yards in the 
area? 


8. How many flagstones 54 ft. long, 3 ft. wide, will be needed 
to lay a sidewalk 1 mile long, 6 ft. wide? 


9. What would it cost, at 10% a square yard, to paint the 
walls of a room 16 ft, 6 in. long, 14 ft. 9 in. wide, 13 ft. 4 in. 
high? 

10. A room 20 ft. long and 17 ft. 6 in. wide will require how 
many yards of carpet 2 ft. 6 in. wide to cover it, making no 
allowance for waste? 


11. Find the weight of a plank 15 ft. 9 in. long, 10 in. wide, 
and 2 in. thick, at 414 lb. per cu. ft. 


12. Find the entire surface of a block of marble 3% ft. long, 
21 ft. wide, and 14 ft. thick. (Draw “ development.” 


MEASUREMENTS. 435 


13. The area of a floor is 1354 sq. ft., and its length is 12 ft. 
8 in. What is its width? 


14. How many bushels will a bin contain, its dimensions 
being 10 ft. 8 in. xX 12 ft. 8in. x 8 ft. 9 in.? (Cancel.) 


15. Find the capacity in gallons of a tank 6 ft. 5 in., by 3 ft. 
9 in., by 4 ft. 6 in. 


16. A farmer wishes to construct a post and rail fence around 
a square field containing 40 acres. He pays 15% each for the 
posts, which are placed 4 rod apart. The rails cost 10 each. 
If the fence is 5 rails high, how much will the material cost? 


17. A farm 1 mile square is divided into square fields each 
containing 40 acres. Make a diagram of the farm, and say how 
many miles of fence will be needed to enclose all the fields. 


(640 A. = 1 sq. mi.) 


18. How many cords of wood in a pile 164 ft. long, 16 ft. 
wide, 30 ft. high? 


19. A cubic foot of water weighs 1,000 oz. What will be the 
weight of a gallon of water? (Give answer in pounds and 
fraction.) 


20. Find the weight of a quart of mercury, considering mer- 
cury 15 times as heavy as water. 


21. A plot of ground 320 ft. long, 210 ft. wide, is enclosed by 
a tight board fence 6 ft. high. How many square yards in the 
surface of the fence ? ARC Acad Ee PES. 


22. Find the number of square yards 
in a sidewalk, six feet wide, on the out- 
side of the above plot. 


! 
' 
! 
! 
! 
( 
| 
| 
| 
! 
| 
| 
| 


016 


The inner rectangle measures 320 x 210 ft. 
What are the dimensions of the outer rectangle? 
Find the difference between the area of the outer rectangle and that of the 
inner one. , 


436 ARITHMETIC. 
23. Find the area of a gravel walk, 6 ft. wide, just inside a 
fence surrounding a plot 320 ft. long, 210 ft. wide. 


Make a diagram. Find difference of areas, as above. 


CAUSE AND EFFECT. 


971. Oral Problems. 
1. If 4 books cost $1.25, what will a dozen cost? 
2. If 8 lb. sugar cost 184, what will be the cost of 50 Ib.? 
3. If 48 lb. tea cost $20, what will 12 lb. cost? 


4. Bought 17 yards of cloth for $30. How many yards 
could I have bought for $90? 


5. If 36 men do a piece of work in 105 days, how long will 
it take 72 men to do it? | 


6. If 7 railway trucks weigh 14 tons, how much would 29 
trucks weigh ? 


7. How long will it take 8 horses to plow a field, if 3 horses 
can do it in 8 days? 


8. What is the height of a steeple that casts a shadow of 
300 ft., if an 8 ft. pole casts a shadow of 12 ft. ? 


9. If 18 men mow 90 acres of grass in 5 days, how many 
acres will 86 men mow in 5 days? In 10 days? 


10. If 60 yd. carpet 2 yd. wide will cover a floor, how many 
yards $ yd. wide will be required? 


972. Slate Problems, 


1. A piece of cloth, measured with a yard measure that is 1 
inch too short, appears to be 25 yd. long. What is its true 
_ length? 

2. Exchanged 40 yd. muslin, worth 101¢ per yd., for 15 yd. 
linen. What is the value of the linen per yd.? | 


CAUSE AND EFFECT. 437 


3. If 3 men or 6 women can do a piece of work in 56 days, 
in what time will 1 man and 2 women working together do it? 


4. If5 mencan do as much in a day as 8 boys, how long will 
it take 32 boys to finish a piece of work which 15 men can do in 
12 days? 

5. If $100 gain $4 in 1 year, what will $350 gain in 3} 
years ? 

6. If 48 horses in 10 days consume 180 bu. oats, how many 
bushels will 82 horses consume in 10 days? In 12 days? In 15 
days? 

7. If 5 men mow 45 acres of grass in 6 days, in how many 
days will 12 men mow 90 acres? 


973. If 5 men mow 45 acres in 6 days, 

1 man will mow 45 acres in 6 x 5 days. 

6x5 
45 


12 men will mow lacre in z, x 5 


1 man will mow 1 acre in 


days. 


12 men will mow 90 acres in 6 x 5 x 90 d 


2 
Canceling, nae = 5. Ans. 5 days. 
2 


974. In practice, the work is somewhat shortened. Since the number of 
days is required, we write the given number of days last, with a line 
underneath. 


6x 5x 90 
iinan mows. Lacra.>—.- 


5 men mow 46 acres days 
12 men mow 90 acres 45 x 12 


If 5 men do the work in a certain time, 1 man will require 5 times as 
many days. We place 5 in the numerator (asa multiplier). To cut 1 acre, 
he will take j. of the time required to cut 45 acres. Place 45 in the 
denominator (as a divisor). 

12 men will take +, of the time 1 man requires. Place 12 in the de- 
nominator. To cut 90 acres will require 90 times as long. Place 90 in 
the numerator. 


438 ARITHMETIC. 


8. If 12 horses eat 60 bushels of oats in 6 days, how many 
bushels will 24 horses eat in 3 days? 


Make bushels the last term. 
12 horses in 6 days eat 


bu. 


l horse in 1 day eats }60 


24 horses in 3 days eat 


975. This example can be solved more easily. 6 days’ food for 12 
horses will supply how many horses for 1 day? 3 days’ food for 24 horses 
will supply how many horses for 1 day? 


9. If 24 men use 240 lb. of beef in 2 weeks, how many 


pounds will 18 men use in 8 weeks? 
24 men in 2 weeks use 240 lb. 


10. If 6 printers can print 1,656 sheets in 9 days, how many 
sheets will 15 printers print in 10 days? 


11. How much will it cost to feed 520 sheep for 36 days, if it 
costs $128 to feed 160 sheep 48 days? 


12. In what time will 8 masons build a wall 84 ft. long, 
working 10 hours a day, if 12 masons build a wall 96 ft. long in 
8 days, working 8 hours a day? 


13. How much money must I lend for 1 year and 3 months, 
when the rate of interest is 5 per cent, in return for $60 lent me 
for 9 months, which I borrowed at 4 per cent? 

14. If 27 men build 54 rods of wall in 6 days, how many 
rods will 32 men build in 9 days? 

15. If 50 men can do a piece of work in 90 days, working 8 
hours a day, in how many days will 72 men do it, working 10 
hours a day ? 

16. If $350 earns $42 interest in 3 years, how much will 
_ $225 earn in 5 years? 


17. Ifa wall 34 feet high could be built by 68 men in 15 days, 
how many men could build a wall 32 feet high in 8 days? 


PARTNERSHIP. 439 


18. Ifa ship’s crew of 500 men have provisions to serve for 
48 days, at the rate of 27 ounces a day for each man, how many 
men will the same provisions serve for 60 days, allowing each 
man 30 ounces a day ? 

. 19. How many hours a day must 9 men work so that they 
may do as much in 16 days as 12 men can do in 16 days of 8 
hours each ? 

20. If 30¢ is paid for 6 lb. 14 oz. of bread, when wheat is 
$1.14 per bu., what should be paid for 23 lb. 12 0z., when wheat 
is $1.32 per bu. ? 

Notr. — Reduce weights to ounces, or to pounds and fractions. 

21. If 3 men can do as much work as 7 boys, how long will 
it take 28 boys to do as much work as 16 men can do in 24 days? 

22. A crew of 16 men have provisions for 36 days, allowing 
20 ounces to each man per day. After sailing 10 days they pick 
up 10 shipwrecked sailors. How long will the provisions then 
last at the daily rate of 16 ounces per man? 


PARTNERSHIP. 


977. Slate Problems. 
1. Band C gain by trade $182. What is the gain of each, 
B having put in $300, and C $400? 
The gain of $700 is $182. Whatshould $300 gain? What should $400 
gain? 
2. A, B,and C invest $720, $340, and $960, respectively. 
The profits are $101. What is each one’s share? 
How many dollars of capital produce $101 profits? 


3. Two men hire a pasture for $45. One puts in 15 cows, 
the other puts in 12 cows. What should each pay ? 


4. A and B hire a boat for 50 days, paying $30. A uses it 
27 days, B uses it 23 days. How much should each pay? 


5. Our standard gold coin consists of 900 parts gold, 90 
parts silver, 10 parts copper. What is the quantity of each 
metal in 50 pounds of coin? 


440 ARITHMETIC. 


6. Gunpowder is composed of 15 parts of saltpeter, 2 of 
sulphur, and 3 of charcoal, mixed together. How many pounds 
of each are there in 72 pounds of powder? 


7. Three farmers hired a threshing-machine for $54. A 
used it to thresh his crop of 900 bu., B to thresh his crop of 828 
bu.; C, 672 bu. How much should each pay ? 


8. A,B, and C rented a warehouse. A stored in it 2,400 
bales cotton; B, 1,500; 0, 1,100. A fire destroyed 1,800 bales. 
How much of the loss should each sustain? 


9. X and Y rent a field for $32. X puts in 8 horses for 
6 months, and Y 10 horses for 8 months. How many dollars 
should each pay ? 


8 horses for 6 months = how many for one month? 


10 horses for 8 months = how many for one month? 


10. M and N entered into partnership. M puts $200 into 
the business for 5 months, and N, $300 for 4 months. They 
gained $132. Find the share of each. 


REVIEW DECIMALS. 


978. Slate Exercises. 
1. Express as decimals 75/5, zygy, and 345. 
2. .3895 + 86.7 + 209.0043 + .81 + 3.075 + 27. 
3. Divide 34,020.072 by 5.309. 
570 + .005 =? 
4. Multiply 80.037 by 10. 
Seventy-three one hundred-thousandths by one hundred. 
.2054 x 1,000 = ? 
5. Subtract 48.8067 from 53.07. 
.0539 x 26.08 =? 


DECIMALS. 441 


6. The smaller of two numbers is 8.5307, and their sum is 
25.07. Find the larger number. 


7. Express .39, 6.175, .00036, and 74.0005 as common frac- 
tions (or mixed numbers). 
8. Divide .826 by 100; 548.71 by 10,000; and fifty-nine ten- 
thousandths by one thousand. 
9. Find the difference between 9.84 and 38.005, and the con- 
tinued product of 83.09, .734, and 5.007. 
10. Reduce 6 shillings 9 pence to the decimal of a pound 
sterling. 
11. Express as decimals seven hundredths, forty-three ten- 
thousandths, and ninety-one millionths. 
12. Change 7, 845, 74s, and s4,, into decimals. Find their 
sum. 
13. Express .42796 as a common fraction, and the sum of 34, 
z3y, and =373, as a decimal. 
14. 3.009 x .07 x .0907. 
15. Divide .0075 by .15, and .00044408 by .0112. 
16. Divisor, 403.6; quotient, 2.709. Dividend? 
.085 x .0056 4 
00007 
18. Change 69 rods to the decimal of a mile. 
19. Change .4285 month (80 days) to days, hours, etc. 


20. How many meters, each 39.37 inches, in 3 miles 220 rods? 


17. What is the value of 


21. Change .1875 bu. to quarts. 

22. What decimal of a pound is 13 oz.? 

23. Reduce 4 ft. 14 in. to the decimal of a rod. 

24. How many links of 7.92 in. each in a 4-rod chain? 

25. A chain is 66 ft. What decimal of an acre is 1 sq. chain? 


449 ARITHMETIC. 


APPROXIMATIONS. 


981. Give approximate answers at sight (Art. 890): 

1. 487% 1s what per cent of 960? 

2. If17 bu. 87 lb. of corn cost $8.75, what will 52 bu. cost? 

3. About how many cords of wood in a pile 25 ft. long, 4 ft. 
wide, 5 ft. high? 

4. How many bushels (14 cu. ft.) can be placed in a bin 6 ft. 
long, 5 ft. wide, 4 ft. high? 

5. How many acres in a field 50 rods long, 80 rods wide? 


6. About how many yards are there in the side of a square 
field containing 1 acre (4,840 sq. yd.) ? 


7. At 74 gal. to cu. ft., about how many gallons will a tank 
hold 6 ft. long, 4 ft. wide, 3 ft. high? 


8. 64.3 + 0987 =? 
9. About how many dollars are equal to £199 17s. 6d.? 
10. A mark = 23.8%. How many marks in $100? 


BONDS AND STOCKS. 


982. Slate Problems. 

1. The people of a certain town wish to build a street rail- 
road. A-company is formed. Five hundred shares of stock, of 
the par value of $100 each, are sold. 

At the end of 6 months it is found that the profits are $2,000. 
How much should the owner of 10 shares receive? 


2. Profits thus distributed are termed dividends. What % 
semi-annual dividend is declared on the stock of the above rail- 
road? ‘To what per cent interest per year is it equal? 


3. Mr. H. has $4,500 in the savings bank, on which he 
receives a low rate of interest. Hearing of the success of the 


BONDS AND STOCKS. 4483 


new road, he gives that amount for 30 shares of the stock. 
What price does he pay per share? What per cent of the par 
value ? 


4. Ifthe semi-annual dividend is again 4%, how much more 
income does Mr. H. receive from the railroad stock than he would 
obtain from the savings bank in six months, interest 4 per cent 
per annum? 


5. What per cent, for six months, does the stock pay on his 
investment of $4,500? What % per year? 


6. If he sells the stock (380 shares) at 1641 (per cent), how 
much more does he receive for it than it cost him? 


7. Which investment will pay better, one in a gas company 
paying 6 per cent dividends annually, their stock selling at 150, 
the other in a bank paying 7 per cent dividends annually, stock 
selling at 175? 


8. What annual dividend should be declared on railroad 
stock bought at 125, so that the buyer will receive 4% per 
annum on his investment? What semi-annual dividend ? 


9. What will be the cost of 17 shares of canal stock, par 
value $50, at 937, and 143 shares gas stock, par value $10, at 
1023 ? 


10. If the above stock is purchased through a broker, what 
commission does the latter receive at 4% on the par value? 


11. A railroad company needing more money to extend its 
road, issues bonds, promising to pay the holder the face value in 
twenty years, with interest at 4%. 

If these bonds are sold at 95, what rate of interest on the money 
invested does the owner of a bond receive? 


12. Government 4 per cent bonds sell for 1164. What ver 
cent interest is received on the amount invested ? 


How is it that these bonds bring higher prices than railroad bonds? 


444 ARITHMETIC. 


13. Can you state a difference between stocks and bonds as to 
the rate of income received from each ? 
Bonds are redeemed at maturity. How about stocks? 


If a railroad prove unsuccessful, which claims are first met, those of the 
stockholders or those of the bondholders? 


14. Why is it necessary sometimes to employ a broker to pur- 
chase stocks or bonds? What is his fee called ? 


15. Mention some other persons, not owners, through whom 
buyers regularly make purchases. 


16. What is the base in the following? 
(a) Insurance; (0) taxes; (c) brokerage; (d) commission; 


(e) interest; (f) discount; (g) stocks; (A) bonds. 


17. At $24.50 per thousand, what will have to be paid in 
taxes by the owner of property assessed at $18,750? 


18. Mr. Cartwright owns a house and lot worth $36,000. 
The tax rate is 21%, and his tax bill is $540. What is the 
assessed value of the property? What per cent of the actual 
value is the assessed value? 


19. If the property in the last problem were assessed at its 
real value, what should be the rate to make he Cartwright’s 
tax bill the same? 


20. For insuring his property, Mr. Cartwright pays a yearly 
premium of $135. If the rate is #%, for how much is his 
property insured? 


21. Reduce 1,674 feet to rods, etc. 


22. A man paid $8,575 for bank stock at 245. How many 
shares, par value $100, did he buy? Ifa quarterly dividend of 
24% is declared, how much should he receive ? 


23. Reduce 7,481 inches to rods, etc. 


24. A woman deposited $100 in a savings bank Jan. 1, 1892. 
On the first of July, interest at the rate of 4% per annum was 


COMPOUND INTEREST. 445 


calculated, and entered on the depositor’s bank book. Jan. 1, 
1893, interest on the new principal was placed to the credit of the 
depositor. The same was done July 1, 18938. How much was 
there to the woman’s credit at the date last mentioned? 


25. Reduce 3,793 feet to rods, etc. 


COMPOUND INTEREST. 


983. Find the amount of $375, for 1 year, at 6%. Consider- 
ing this as a new principal, find the amount for a year, same 
rate. Find the amount of this last principal for 3 months. 

26. What is the amount of $375, for 2 years 3 months, at 
6%, compound interest? 


27. What is the amount of $375, for 2 years 3 months, at 6%, 
the interest compounded semi-annually ? 
Principal, $375. 

3% 11.25 6 months’ interest. 
386.25 Amount 6 months. 
3% 11.5875 6 months’ interest. 

Amount 1 year. 
3% 6 months’ interest. 
Amount 1} years. 

etc., etc., etc. 


Find the “compound interest”’ on $375, for 2 years 3 months, 
at 6 per cent, compounded semi-annually. 
28. What is the amount of $100, at compound interest, for 3 


years, interest at 6%, compounded annually? 


29. Find the compound interest of $1,800, at 4%, for 2 years, 
interest compounded quarterly. : 


$ 1,800.00 

1%, 18,00 
1,818.00 

1%, 18.18 


etc., etc., etc. 


446 ARITHMETIC. 


30. Find the difference between the simple interest of $100, 
for 2 yr. 3 mo., at 5%, and the compound interest for the 
same time, interest compounded semi-annually. 


$ 100.00 . 
22%=25 2.50 Divide by 4, and put first quotient figure 
102.50 one place to the right. 
24% 2.5625 
105.0625 
21 97 2.6266 


$ 107.6891 (four places of decimals are sufficient.) 


984. Compound interest is allowed by savings banks. It is 
not collectible on notes or other debts. 


REVIEW. 
985. Oral Problems. 
1. A capitalist wishes to realize 5% on money invested in 


stock. What must be the annual dividend on stock costing 300, 
in order to produce this rate? 


2. What will be the taxes on property assessed at $25,000, 
the rate being $16 per $1,000? 
3. Find the compound interest on $1,000, for two years, at 


five per cent, interest compounded annually. 


4. What will be the net cost of an article marked $8, on 
which a discount of 50, 25, and 10% is allowed? 


5. Find the “list” price of an article sold for $10 after a 
discount of 50 and 50 per cent had been deducted. 


6. Paid 90¢ for an article. The discount is 25 and 25 per 
cent. What is the list price? 


7. One boy can do a certain piece of work in 2 hours, a 
second boy requires 3 hours, a third needs 6 hours. How long 
will it take the three working together? 


8. Sold a cow for $60, losing 25%. What was the loss? 
9. Sold a cow for $60, gaining 25%. What was the gain? 


EXCHANGE. 447 


10. Sold two horses at $240 apiece. On one I gained 20%, 
on the other I lost 20%. Did I gain or lose on both, and how 
much ? 


Suaaestion. — $ 240 in the first case represents 120% of cost of horse. 
The gain is 20%, which is 4 of selling price, or $40. 

The loss in the other case is 20%, which is what part of the selling price ? 
Do not find the cost. 


11. John has $60, James has $80. James has what per cent 
more money than John? John has what per cent less money 
than James? 


12. 2is what per cent of 4? 41s what per cent of 4? 


13. Two men working together can finish a piece of work in 
8 days; one can doit in 12 days. How long would the other 
take to do the work? 


14. How many yards of cloth at $3.75 per yard can be 
bought for $90? 


EXCHANGE. 


992. If I wish to pay a bill in a distant city, ought I to enclose the 
money in a letter? Why? 

Can money be sent by express? Can the telegraph be used in paying 
money at a distance? 


What is a money-order ? 

Can I buy from the postal authorities a money-order payable in Europe? 
What will be the cost of a money-order for $ 85, payable in San Francisco ? 
What is the largest money-order that can be purchased? 


What is a check? Can you tell why a draft rather than a check is used 
in paying a bill at a distance? 


Pupils should be encouraged to look up answers to the fore- 
going. 


448 ARITHMETIC. 


Bills of exchange are either foreign or domestic. A domestic 
bill of exchange is called a draft, the term bil of exchange being 
generally applied only to foreign bills. 


DOMESTIC EXCHANGE. 


993. Slate Problems. 


William F. Smith, of Memphis, Tenn., owes John M. Thomson, 
of New York, $3,475.86. He purchases from a Memphis banker, 
Joseph E. Washington, a sight draft for the above amount on the 
Chemical Bank of New York. The following is the form of the 
draft: 


$3,475,85,. Mempuis, Tenn., Aug. 9, 1893.' 


At sight, pay to the order of John M. Thomson Three 
Thousand Four Hundred Seventy-five and 88 Dollars, value 
_ received, and charge to the account of 


To CuEMiIcAL BANK, JOSEPH EK. WASHINGTON. 
New York. 


1. What must William F. Smith pay for the above draft, the 
rate being $1.50 premium per $1,000? 


(A draft for $1,000 costs $1,001.50.) 


2. Find the cost of a Boston draft on New York for $1,875, 
at 12¢ discount per $1,000. 


(A draft for $1,000 costs $999.88.) 


3. What will a St. Louis merchant have to pay for a draft 
on New York for $2,460.53, at 50% premium per $1,000? 


4. If the rate of exchange is 50% discount per $1,000, what 
is the face of the sight draft on Boston, that can be bought in 
New York for $1,000? 


5. When the premium is $1.25 per $1,000, Mr. Brown pays 
$1,634.04 for a draft on Louisville. What is the face of the 
draft? 


REVIEW. 449 


6. At4% premium, find the cost of a sight draft for $ 1,843.60. 


$ 1,843.60 
Aches 2.30 Add. 


7. At 75¢ discount per $1,000, how much will cost a sight 
draft on Milwaukee for $946.75? 


$946.75 
50 perM. 473 


as 25 per M. 237 


8. Paid $632.18 for a sight draft on Milwaukee. What was 
the face of the draft, the discount being 3,%? | 


9. I sent a commission merchant $1,000 to buy grain. How 
much will he spend for grain, if his commission at 14% is 
included in the amount sent? 


(Let « = amount spent for grain. ? = commission.) 


_ 10. A farmer ships produce to a commission merchant, which 
the latter sells for $339.66, charging 2 per cent commission. For 
the remainder of the money he buys groceries and dry-goods, 
charging 2 per cent commission on the amount spent. What is 
the cost of the goods purchased ? 


REVIEW. 
994. Slate Problems. 


1. A joiner worked on Monday 9 hr. 45 min., on Tuesday 
and Wednesday 10 hr. 45 min. each day, on Thursday and Fri- 
day 10 hr. 15 min. each day, and on Saturday 6 hr. 45 min. 
What was the average length of his day’s work? 


2. A watch that loses 85 seconds in an hour was set right 
at noon on Monday. What time did it show at 6 p.m. the fol- 
lowing Thursday ? 

3. There are 5 boys whose heights are 4 ft. 9 in., 5 ft. 1 in., 
4 ft. 5in., 3 ft. 11 in., and 4 ft. 4 in., respectively. What is 
their average height ? 


450 ARITHMETIC. 


4. A man hada plot of ground 20 yards long and 12 yards 
wide, which he planted in cabbage. How many plants did he 
require, if the rows, which ran lengthwise, were 2 feet apart and 
2 feet from the fence surrounding the plot, and the plants in the 
rows 16 inches from each other and from the fence ? 


Get the correct number of rows, and the correct number of plants in a 
row. How many plants would have been needed if the rows ran crosswise ? 


5. How long would it take a person to count a million silver 
dollars, at the rate of 100 a minute, and working 8 hours a day? 


6. The front wheel of a wagon is 13 ft. 4 in. in circumference. 
How many revolutions will it make in a journey of 14 miles? 
How many more revolutions will it make than the hind wheel, 
the circumference of the latter being 17 ft. 6 in.? 


7. The wheels of an engine being 16 ft. 8 in. in circumfer- 
ence, and the number of revolutions 150 per minute, how far 
does it goin an hour? Give answer in miles and rods. 


995. Circular Measure. 


60 seconds (/7) —-1 minute (’) 
60 minutes 1 degree (°) 
360 degrees 1 circle. 


8. If the equatorial circumference of the earth is 25,000 
miles, how many miles apart are two places on the equator, the 
distance between them being 20°? 


9. What is the length of a degree on a circle whose diameter 
is 18 feet? 
The circumference = diameter x 3.1416. 

10. The 60th parallel of latitude is a circle about one-half as 
long as the equator. How far due east of Christiania is St. 
Petersburg, both situated on this parallel, the former being 10° 
east of Greenwich, and the latter 30° east ? 


11. How many miles north of the equator is a place in lati- 
tude 46° 22' 30"? Take 694 miles to a degree, 


EXCHANGE. 451 


12. Two places in latitude 45° are 22° 30! apart, measured on 
that parallel. Find the distance in miles, assuming the 45th 
parallel to be a circle .7071 times the length of the equator, and 
considering the length of the latter to be 25,000 miles. 


996. Time Drafts. 


$ 9875875. New Orxrans, June 15, 1893. 

At three days’ sight, pay to the order of John D. Hallock, 
Nine Hundred Eighty-seven 55>, Dollars, value received, and 
charge to account of 


To Natronat Park Bank, FRANK PHILLIPS. 


New York. 


When Mr. Hallock receives the above, he presents it to the National 
Park Bank for acceptance. The proper bank official writes across the face 
of the draft in red ink “ Accepted,” with the date, say ‘“‘ June 18, 1893,” and 
signs his name. Three days thereafter, plus three days of grace, or June 24, 
the draft will be payable. 


997. Sight drafts are usually not allowed days of grace. Time drafts 
are generally allowed three days of grace. (See Appendix.) 


998. The premium on the above draft at $1.50 per $1,000 is calcu- 
lated on the face of the draft, and amounts to $1.48. 


999. Since it is not payable until six days after acceptance, the inter- 
est (or bank discount) for that time is deducted. 


Interest on $987,565, for 6 days at 6% = $.99. 
Cost of draft = $987.65 + $1.48 — $.99 = $988.14. 


N.B. Take 6% as the interest rate, unless a different rate be expressed. 


1000. Slate Exercises. 


1. What will I have to pay for a 90-days draft on San Fran- 
cisco for $840, at $1.75 premium per $1,000? 


2. Face $400; 30 days’ sight; discount 4%. Cost? 


3. Face $560; 60 days’ sight; premium 50 per $1,000. 
Find cost. 


4.52 ARITHMETIC. 


4. What will be the cost of a sight draft for $ 625.38 at 71 ¢ 
discount per $1,000? 


_ 5. Find the cost of a 60-days draft for x dollars, premium 
25 ¢ per $1,000. 


6. Find the cost of an z-day draft for $ 1,200, discount 4%. 


7. Find the cost of a 30-days draft for $1,600, premium 
x dollars per $1,000. 


8. Paid $1,188.90 for a 60-days draft, at 1% premium. 
What was the face of the draft ? 


9. A time draft for $1,800 at $1 premium per $1,000, cost 
$1791.90. At how many days’ sight was it drawn? 


10. At what rate did I purchase a 90-days draft for $900, its 
cost being $884.70? 


LONGITUDE AND TIME. 


Notr.— This topic should be taught in connection with the study of 
Mathematical Geography. The globe should be used to show the pupils 
that all places on the same meridian have the same time, that a difference 
in longitude of 15 degrees produces a difference in time of 1 hour, and that 
the more easterly of two places has the later time. | 


1001. Oral Problems. 


1. The difference in time being 1 hour for each 15 degrees, 
find the difference in longitude between two cities differing in 
time 34 hours. 


2. Two places differ in longitude 61 degrees. What is their 
difference in time? 


3. London is 75° east of Philadelphia. When it is 1 o’clock 
at Philadelphia, what is the time at London? 


4. When it is 2 p.m. at London, what is the time at Phila- 
delphia ? 


—— 


LONGITUDE AND TIME. 453 


5. When it is noon at a city 25 degrees west of Vienna, what 
is the time at the latter place? 


6. How many degrees of longitude correspond to a time dif- 
ference of 3 hours 40 minutes? 


7. What is the difference in longitude between Philadelphia, 
75° west longitude, and St. Petersburg, 30° east longitude ? 


8. When it is 3 p.m. at St. Petersburg, what is the time at 
Philadelphia ? 


9. Washington is in 77° west longitude, and uses “standard 
time,”’ that is, the time of 75° west longitude. What is the 
difference between the correct time at Washington and its clock 
time ? 


10. A town in 84° west longitude uses standard time (of 90°). 
What is the correct time when the clocks are striking 12, noon? 


1002. Slate Problems. 


1. Find the difference in longitude between two places dif- 
fering in time 3 hr. 44 min. 


2. Two places differ in longitude 37°18’. What is their 


difference in time? 


3. Chicago is 87° 35! west of Greenwich. What is the dif- 
ference in time between the two places? 
Is it earlier. or later than noon at Chicago when it.is noon at 
Greenwich? Why? 
What is the standard time at Chicago when it is 1 P.M. at 
Greenwich? 


4. When a captain’s observation of the sun shows that it is 
exactly noon, the ship’s chronometer, keeping Greenwich time, 
reads 30 minutes past 2p.m. How many degrees west of Green- 
wich is the vessel ? 


5. Find the difference in time between two places in longi- 
tude 74° 31' and 93° 14! west of Greenwich, respectively. 


454 ARITHMETIC. 


6. When it is noon at a place 11° east of Greenwich, it is 
1.30 p.m. at another place. Find the longitude of the latter place. 

7. A train ran from New York to San Francisco, 3,313.5 
miles, in 3 da. 12 hr. 17 min. How many miles per hour did it 
average ? 

8. If for $6 I can have 1,200 pounds carried 36 miles, how 
many pounds can I have carried 24 miles for the same money ? 


9. At 80% per ounce, what is the value of 86 ingots of 
silver, each weighing 2 lb. 10 oz. 15 pwt.? 


10. Find 30 per cent of 27 yards 8 inches. 


11. The solid contents of a block 12 feet 6 inches wide and 3 
feet 9 inches thick are 27 cubic yards 1 cubic foot 810 cubic 
inches. Required its length. 


12. A farmer sold 237 bushels 3 pecks of wheat, which was 48 


per cent of his crop. How many bushels, pecks, etc., did he 
have left? 

(48% is given; you have to find what %? What part of 48% added to 
itself will give the required per cent? Do not find the whole crop.) 


13. How many spoons, each weighing 2 ounces 12 penny- 
weights, can be made from 4 pounds 4 ounces of silver? 

14. A man travels due west, on the 45th parallel of latitude, 
84 miles per hour for 24 hours. How many degrees has he 
traveled, the length of a degree being 48.96 miles? 


REVIEW. 
1003. Oral Problems. 


1. A puts $600 into business; B, $400; the profits are $500. 
What is the share of each ? 


2. Two boys hire a camera for 26 weeks, paying $5.20. 
How much should be paid by the boy that uses it 12 weeks? 


3. New Orleans is 90° west of Greenwich. When it is 2 P.M. 
at the latter place, what is the time at New Orleans? 


EXCHANGE. 455 


4. Find the discount, at 6%, on a note for $300, that has 48 
days to run. 


5. What will be the cost of 84 yards of muslin at 49% a yard? 


6. Two men hire a pasture for $84. One puts in twice as 
many head of cattle as the other. What should each pay? 


BILLS OF EXCHANGE. 


Exchange for £180 17s. 6d. New York, Dec. 14, 1895. 
Sixty days after sight of this First of Exchange (Second 
unpaid), pay to the order of John W. Moran & Bro., One Hun- 
dred Highty pounds sterling, seventeen shillings, six pence, 
Value received, and charge the same to account of 


To JAMES Lennon & Co., 
London. 
No. 39. 


PETER COMERFORD & Son. 


1005. Slate Exercises. 


1. Find the cost of the above bill at $4.87 per &. 


£200 = $974.00 
20 = 
£180=§ 
10s. = 2.435 £4 
5s; = 
2s. 6d. = 


oy 


$ 
2. What would be the cost of a cable transfer of £251 11s. 
9d., at $4.881 per £? 
£250 = $1,221.25 +1 of £1,000 

l= 

10s. = 

1s. = 

6d. = 

3s. = 


The newspapers give quotations of foreign exchange for sight and 60-day 
bills, also for cable transfers. 


456 ARITHMETIC. 


-_ 


1006. The New York quotations for French exchange give the number 


of francs for $1. i, 
Paris cable transfers 5.164 @ 5.153. 


Paris bankers’ 60 days 5.183 @ 5.18}. 
Paris bankers’ sight 5.163 @ 5.164. 


1007. The quotations for German exchange give the value in U.S. 
money of 4 Reichmarks (or marks). 


Reichmarks (4) 60 days 954 @ 95}. 
Reichmarks (4) sight 953 @ 954. 
3. Find the cost of a sight bill on Paris for 1,000 frances, at 
5.164 francs for $1. 
4. Find the cost of a 60-days bill of exchange on Berlin for 
1,874.85 marks, at 951¢ for 4 marks. 
5. What will be the face in marks of a sight bill of exchange 
on Berlin that can be bought for $1,000, at 9519 for 4 marks? 


6. A New York merchant pays $1,637.50 for a 60-days bill 
on Paris. What is the face of the bill, the rate of exchange 
being 5.183 francs for $1? 


7. At $4.88 per £, what will be the face of the cee bill on 
London that can be bought for $1,500? 


18750 £307 s., etc. 
1399.99 _ 18750 61)18750 
ABB 61 450 
61 £23 remainder 
20 


460s., new dividend. 


8. Bought goods in London amounting to £437 5s. 10d. less 
4%. How much will I have to pay in Boston for a sight bill of 
exchange at $4.884, to settle the account? 


9. What will be the cost in Chicago for a 60-day bill on Paris, 

that will pay for the following articles? Rate, 1 franc = 192. 
18 pieces silk, 44 meters each, at 25 francs per meter, less 74%. 
3 pieces of cloth, 50 meters each, at 20 francs per meter, less 5%. 
Packing charges, 60.50 francs. 


EXCHANGE. | 457 


10. I wish to send a sight bill of exchange on Berlin in pay- 
ment of the following invoice : 

4 cases musical instruments amounting to 3,598.60 marks, less 
10, 5, and 24%. 

Freight to Hamburg, 165 kilos, at 4.80 marks per kilo. 

At 95£¢ for 4 marks, what will be the cost of the bill of 
exchange ? 


CHAPTER XIII. 


PARTIAL PAYMENTS. — RATIO AND PROPORTION. — SQUARE 
ROOT. —SURFACES AND VOLUMES. 


PARTIAL PAYMENTS. 


1008. U.S. Rule. 
DututH, Mrinn., Jan. 5, 1889. 
On demand, I promise to pay to the order of Owen McGee 
Three Hundred Dollars, value received, with interest at 7 per 
cent. 
$3005. J. RANDOLPH PAGE. 


Endorsements: May 20, 1889, $100; Oct. 30, 1889, $100; 
March 6, 1890, $50. 


How much was due Jan. 5, 1891? 


Find amount of $300 Jan. 5, 1889, to first payment May 20, 1889, 4 mo. 


15 da. (by compound subtraction). $307.88 
Deduct first payment, 100.00 
Balance May 20, 1889, $ 207.88 
Interest on $ 207.88 to Oct. 30, 5 mo. 10 da., 6.47 | 
Amount, $ 214.35 
Less second payment, 100.00 
Balance Oct. 30, 1889, $ 114.35 
Interest on $114.35 Oct. 30 to March 6,4 mo.6da., ° 2.80 
Amount, $117.15 
Less third payment, 50.00 
Balance March 6, 1890, $67.15 
Interest on $67.15 March 6 toJan. 5,9 mo. 29 da, —_—3.90 
Due Jan. 5, 1891, $71.05 


458 


PARTIAL PAYMENTS, 459 


1009. Slate Exercises. 


Norr.— Find time by compound subtraction. 


1. How much is due June 8, 1896, on a demand note for 
$1,200, with interest at 6%, dated June 3, 1898, bearing en- 
dorsements of payment of $500, Sept. 18, 1894; $600, Jan. 38, 
1895? 


2. A demand note for $600, bearing interest at 5%, was 
given Feb. 18, 1892. A payment of $250 was made May 28, 
1898; one of $150 was made Oct. 8, 1898. How much is due 
Jan. 23, 1895? 


3. Note for $2,000; interest, 7%; dated April 15, 1891. 
Endorsements : $50, Sept. 20, 1891; $100, May 26, 1892; $1,000, 
June 20, 1893. How much is due Dec. 27, 1894? 


1010. Face of note, $ 2,000.00 
Interest from April 15 to Sept. 20, 1891, 5 mo. 5 da., ; 60.28 
Amount due Sept. 20, 1891, $ 2,060.28 


If the $50 payment were deducted, and interest computed on 

the balance, $2,010.27, the maker would be charged interest on 

$10.27 more than the face of the note, and this the law does not 

allow. Interest is taken on $2,000 until next payment, May 

26, 1892, 8 mo. 6 da., 95.67 
Amount due May 26, 1892, $ 2,155.95 
As the two payments are not large enough to meet the interest 

now due, the interest is again calculated on the original $2,000 


from May 26, 1892, to June 20, 1893, 1 yr. 24 da., 149.33 
Amount, $ 2,305.28 
Less $50 + $100 + $1,000 (three payments), 1,150.00 
Balance due June 20, 1893, $1,155.28 
Interest on $1,155.26 to Dec. 27, 1894, 1 yr. 6 mo. 7 da., 122.87 
Due Dec. 27, 1894, $ 1,278.15 


1011. By the United States rule for partial payments, the 
amount of the principal is found to the time when the payment, or 
the sum of two or more payments, equals or exceeds the interest. 

From this amount deduct the payment or sum of payments. 

Use the balance then due as a new principal, and proceed as 
before, 


460 ARITHMETIC. 


4, ApBany, N.Y., March 5, 1893. 

One year after date, I promise to pay John Harrigan, or 
order, Nine Hundred Dollars, value received, with interest at six 
per cent. 

$ 9005%%5 ANDREW T. SULLIVAN. 


Endorsed as follows: June 5, 1893, $10; Sept. 5, 1893, $50 ; 
Jan. 5, 1894, $120. 


What was due March 8, 1894? 


1012. In the United States courts, and in those of some of the states, 
interest for a portion of a year is taken by days, upon the basis of 365 days 
to the year. 

To make the work easier for the pupils, however, the year of 360 days 
should be used in the examples given, and the time between dates should 
be found by compound subtraction. 


PRESENT WORTH AND TRUE DISCOUNT. 


1016. Problems are frequently met with in books, in which the “ pres- 
ent worth” is asked of a sum of money payable at a future date. 


1017. 1. What is the present worth of $150 payable in 1 
year 6 months, interest 6% ? 
By this is meant what sum at 6 % interest will amount to $ 150 in 1 year 


6 months? Or, 
Given the amount ($150), rate 6%, time 1 yr. 6 mo., to find principal. 


x + (a X 85 X 14) = 160. 


1018. By “true discount” is meant the difference between the sum 
payable at a future time and its ‘present worth.” 


2. What is the “true discount’ of $150, payable in 1 year 
6 months, interest 6% ? 


The amount $150, rate 6%, time 1 yr. 6 mo., are given. Find the 
interest. 
Let # = principal 
Amount = @ + (% X 785 X 14) = 150 
Interest = amount — x 


MENSURATION, 461 


SURFACES AND VOLUMES. 


1024. Slate Problems. 

1. If a piece of cloth is 20 yards long and 3 yd. broad, how 
broad is another piece of cloth 12 yards long ay, contains as 
many square yards as the former? 

2. An iron beam 16 ft. long, 24 ft. broad, and 8 in. thick, 
weighs 1,280 lb. What is the length of a similar beam whose 
breadth is 34 ft., thickness 74 in., and weight 2,028 lb. ? 


3. What will it cost to carpet a room 224 ft. long by 153 ft. 
wide with carpet 24 ft. wide, costing $1.50 per yd.? 

4. What is the length of a box 62 ft. wide and 74 ft. high, 
that will exactly contain 12 boxes 4% ft. long, 34 ft. wide, and 
23 ft. deep? _ 

5. What is the value, at $120 per acre, of a square field 
whose side 1s 35.25 chains ? 


10 sq. chains = 1 acre. 


6. What is the area in square feet of a triangle whose base 
is 18 ft. 4 in., and whose altitude is 11 ft. 10 in.? 

7. What is the area of a circle whose diameter is 7.5 feet, 
the area of the circle being .7854 times the area of the square 
that will just enclose it? 

8. Find the capacity, in bushels, of a bin 22 ft. ahs 14 ft. 
wide, 12 ft. high? 


9. How many gallons will a tank hold, its dimensions being 
4 ft. lin. by 3 ft. 8 in. by 2 ft. 3 in.? 


10. How many square yards are there in the walls and the 
ceiling of a room 21 ft. long, 18 ft. wide, 12 ft. high? Make a 
diagram. 

11. A tank 54 ft. by 6 ft. by 7 ft. can be emptied by two 
pipes, one of which discharges 9 gallons per minute and the 


462 ARITHMETIC. 


other 7 gallons per minute. How long will it take each to 
empty the tank? How long will it take both together? 


12. A parlor is 18 feet long, 15 feet wide. Make a diagram 
showing how carpet 27 inches wide can be laid without cutting 
the carpet lengthwise. Which would be the better way to lay 
carpet 30 inches wide in the above room ? 


13. Calculate the number of running yards of carpet 30 in. 
wide needed for the floor of the above room, including 44 yards 
wasted in matching the pattern. 

Find the cost of carpeting the room at 95 cents per running 
yard for carpet, 5 cents per square yard for lining, and 10 cents 
per running yard for sewing and laying. 


14. A rug 18 feet long, 15 feet wide, is placed in the centre 
of the floor of a room 21 feet long, 18 feet wide. What is the 
width of the strip left uncovered? Find the area of the uncov- 
ered space ? 


15. A room is 18 feet wide, 24 feet long, 9 feet high. There 
are two doors 4 feet wide, 74 feet high; two windows 4 feet wide, 
6 feet high; and a fire-place 5 feet square. How many square 
feet of plastering will there be on the walls and ceiling, deduct- 
ing for a baseboard 12 inches wide? How many running feet of 
baseboard will be needed ? 

Draw “development” of the above room, showing the four 
walls and the ceiling, and locating the doors, the windows, and 
the baseboard. 


Do not use baseboard where it is not required. 


16..At the rate of $1,400 for a pile of lumber 25 ft. long, 20 
ft. wide, 10 ft. high, what is the value of a pile 50 ft. long, 40 
ft. wide, 20 ft. high? 


17. Ifit costs $14 to paint the walls and the ceiling of a room 
25 ft. long, 20 ft. wide, and 10 ft. high, what will it cost to paint 
the walls and the ceiling of a room 50 ft. long, 40 ft. wide. and 
20 ft. high? ) 


INVOLUTION. 4638 


SQUARE ROOT. 

1029. Squaring a number is multiplying the number by itself. 
The square of 8 = 8 X 8= 64. 

1030. The square of a number is indicated by writing a small 

2a little to the right of the upper part of the number. 
5? = 25, 12? = 144. 
What is the square of 4? Of 6? Of 7? Of 9? Of 10? Of 11? 
2=? B=? 
Square 13. 15. 21. 16. 19. 1#@=? 17?=? 24°=? 33?=? 


1O31. The square of 25 = (20+ 5) x (20+ 5). 


20 +6 

20 +5 
Multiplying by 20 207+ 20x5 
Multiplying by 5 20x 5 +5 


202 + 2(20 x 5) + 5? = 400 + 200 + 25 = 625. 


1032. The square of the sum of two numbers is equal to the 
square of the first + twice the product of the first by the second 
+ the square of the second. 


13?=(10+38)?= 10°+2(10x3)+3?=? 
18*= (10 +8)? = 100 +1604 64=? 
O71? = (20+ 7)? = 400 +2804 49 =? 


1033. Oral Exercises. 


Square : 

1. 14 4. 22 Typ OL 10. 32 13. 24 

pei ta, 5. 31 8. 61 11. 42 14. 33 

Be EMA 6. 41 9. 23 12. 52 15. 43 

1034. The square root of 41s 2; of 9is 8; of 16 is 4; of 25 
is 5. 


1035. Give the square root of 86. Of 64. Of 81. Of 121. 
Of49. Of 100. Of 144. 


464 ARITHMETIC. 


1036. The sign of square root is +/. 
V8 0. VIEL ei Veo ny oes 


1037. Find the square root of 169. 


10?= 100. 20?=400. The square root is between 10 and 20; it is, 
therefore, 10 + a second number. 
169 = 10? + 2(10 x second) + second?. 
169 = 100 + 20 x second + second ?. 
20 x second + second? = 69. 
From this it appears that the second number is 3, since 
20 x 3 + 3? = 69. 


1038. It may be shown in this way: : 
_10 (first number) 


169 
10? = 100 
Trial divisor — twice 10 20)  69(3 second number) 
60 
9 
3= 9 
Ans. 10 +3 = 13. 


1039. Find the square root of 2,116. 
40 (first number) 


2,116 
40° 1,600 
40 x 2 = 80, trial divisor ) 516(6 second number) 
480 
36 = 6? 


Ans. 46. 


1040. Instead of multiplying the trial divisor by the second number, 
and then ascertaining whether the remainder is the square of the second 
number, the second number is added to the trial divisor and this sum is mul- 


tiplied by the second number. 
40 (first number) 


2,116 
1,600 
(2 x 40) +6 = 86) 516(6 second number) 
516 
Ans. 46. 


REVIEW. 465 


1041. In practice, the work is shortened by omitting the ciphers. 
First, point off in periods of two figures, commenc- AWA 
ing at the right. Find the greatest square in the first on org 
period, and place the root in the quotient. Subtract 21'16 
the square from the first period. Bring down the next 16 
period. Multiply the first quotient figure by 2,and 86) 516 
use it as a trial divisor. Place the second figure in 516 
the quotient. Affix it also to the trial divisor. Mul- (ian 
tiply the two figures in the trial divisor by the second quotient figure. 


1042. Slate Exercises, 
Extract the square root: 


ee boo 6. 1,296 11. 2,809 16. 5,625 
2. 256 7. 1,225 127)72:916 pe AG etet: 
3. 324 8. 1,764 13. 3,721 18. 7,056 
4. 576 9. 1,936 14. 3,969 19. 8,281 
5. 676 10. 2,601 15. 5,184 20. 9,025 


REVIEW. 
1043. Slate Exercises. 


Divide (Arts. 385, 616): 
1. 4,270,978,096 + 564,347 
4,3875,621,423 + 856,789 


2,171,008,895 - 721,985 
. 86,409,429,120 + 876,008 
4,518,821,072 + 752,134 . 57,681,954,968 + 768,437 
8,817 832,184 ~~ 607,432 . 40,333,410,989 + 568,709 
8,462,706,614 = 567,843 10. 58,531,676,960 + 678,432 


oO PR WO ND 
oanrt 


Write answers (Art. 385): 


1, 450,000 4, 700,000 901,020 
86,432 59,084 98,642 
, 500,000 5. 683,427 385,098 
72,356 67,805 76,057 
588,217 701,380 673,217 
64,587 58,437 85,607 


466 ARITHMETIC. 


1044. An Invoice (English). 


Invoice of 3 bales Linen Goods forwarded by rail to Glasgow, 
for shipment thence per 8.8. Anchoria to New York, to order, 
and for account and risk of Messrs. Robinson & Oo. 


yd. 
[R]Co. | #2 | 30 pes. Bord. Crash 1500 | 1g jj £11 | 14 [43 
ag) peaked “iahhs 1500 | 2 « 
#3 | 60 “ Checked G. C. 3000 | 15° 
#4 | 60 “ ms « 2889 | 253 « A 
£ 
Less 24% disct. aut A 
£ 


1. Find the duty in U.S. money at 50% ad valorem. 
£ =$4.8665. 


2. What is the cost in English money of crockery amounting 
to £166 18s. 4d. less a discount of 5 and 5% ? 


RATIO. 


1045. Ratio is the relation which one number has to another 
of the same kind. 


1046. The first term of the ratio is called the antecedent - the 
second, the consequent. 


The ratio of 3 to 6, $9 to $18, 15 cows to 30 cows may be 
expressed 2, 7%, 43. They are each equal to 4. 


1047. Oral Exercises. 
lixpress the ratio in lowest terms: 


1. 275 On se 4. 3 quarts to 4 gallons 


AL Seis Notr. — The denominations must 


2. $19 to $95 be the same. 
3. $36.50 to $18.25 3 quarts to 16 quarts = is 


RATIO. 467 


5. 6 pecks to 5 bushels 8. 1 gallon to 500 cu. in. 


6. 20 mills to 1 dollar 


7. 7 tenths to 3 fifths 10. 1 shilling (24.33) to 1 dollar 


1048. Sight Exercises, 


9. 1 mark (23.8) to 1 france (19.3 ¢) 


fs Be: Bi Leica 
Leia G4 nies Oo ae hae 
18 386 PS R36 
Yaak a = 
OG wee, rane ew) 
i138 165 PA yo 
Peyaaels LAE og  Plb 2 imarks’ 
: 3bu. $24 ie? 21 marks 
3 BS qt. _ 30% 10. 5+22—?+88 
lgal. ?¢ 


11. 6 horses + ? horses = $600 + $900 
12. 1 ft.+? yd. =15¢+ 90% 
13. 1 qt. 1 pt. +1 pt. =?¢+4¢ 


1049. Oral Problems, 


1. One line is a rod long, another is 53 ft. long. What is the 
ratio of the first to the second ? 


2. What is the ratio of 7 hours to 1 day? 


3. A pound of coffee costs 30%, of sugar 6%. What is the 
ratio of their respective prices? 


4. A walks in 4 hours as far as B in 5. What is the ratio 
of A’s speed to B’s? 


468 ARITHMETIC. 


5. HE earns in 6 days as much as D earns in 8 days. Find 
the ratio of H’s daily earnings to D’s. 

6. One wheel makes 800 revolutions in 2 minutes, the 
second requires only 1} minutes to make the same number. 
Find the ratio of the number of revolutions made by the first 
wheel in 1 minute to the number made by the second wheel in 
the same time. 

7. A circle whose diameter is 1 ft. has a circumference of 34 
ft. What is the ratio of the diameter to the circumference? 


8. One train goes 40 miles an hour, a second goes 45 miles 
an hour. What is the ratio of the speed of the first to that of 
the second ? 

9. A window is 6 ft. 4 in. high by 4 ft. 2in. wide. What is 
the ratio of the height to the width? 

10. A father is 36 years old, his son is 9. What was the 
ratio 6 years ago of the father’s age to that of the son? 


1050. Slate Problems. 
(Be sure your answer is correct.) 

1. One line is 3 rods 4 yards long, the length of another is 
5 rods 1 ft. Find the ratio of the first to the second. 

2. One candle lasts 4 hours 20 minutes, another lasts 3 hours 
15 minutes. Find the ratio of the first to the second. 

3. A pound of coffee costs 2549; 1 lb. of sugar costs 5,3, 
What is the ratio of price of sugar to that of coffee? 

4. M walks in 1 hour 47 min. as far as N walks in 2 hours 
3 minutes. What is the ratio of M’s speed to N’s? 

5. P earns in 19$ days as much as Q in 183 days. What is 
the ratio of Q’s daily earnings to P’s? Of P’s to Q’s? 


6. One wheel makes 600 revolutions in 84 seconds; a second 
makes 300 revolutions in 84 seconds. What is the ratio of the 
speed of the first wheel to that of the second? 


INTEREST AND DISCOUNT. 469 


7. The circumference of a circle is 12.5664 ft., and its radius 
is 2 ft. What is the ratio of the diameter to the circumference ? 


8. One train goes 40 miles in 50 minutes, another goes 24 
miles in a half hour. What is the ratio of the speed of the sec- 
ond to that of the first? 


9. One window is 6 ft. 8 in. x 4 ft. 2 in.; a-second is 4 ft. 8 
in. x 2 ft. lin. What is the ratio of the area of the second to 
that of the first? 


10. A mother is now 35 years old, and her son is 3 years and 
6 months old. Fourteen months ago, what was the ratio of the 
mother’s age to that of her son? 


11. A farm costing $4,750 was sold for $5,750. What is the 
ratio between the profit and the cost? 


12. A man can do a piece of work in 42 days. What part of 
it can he do in a day and a half? What decimal? What per 
cent? 


13. What is the ratio between a ton of 2,000 pounds and one 
of 2,240 pounds? 


INTEREST AND DISCOUNT. 


1052. Slate Exercises. 
(Solve the first ten by aliquot parts.) 


Find the amount: 
1. $1,875.25, 3 yr. 5 mo. 15 da., 44%. 
2. $487.50, 1 yr. 10 mo. 25 da., 6%. 
3. $1,206.84, 2 yr. 1 mo. 16 da., 5%. 
4. $595.00, 7 yr. 7 mo. 7 da., 7%. 
7mo.=7y of 7 yr. 7 da. = what part of 7 mo. 


470 


ARITHMETIC. 


$763.25, 8 mo. 11 da., 4%. 
$685.70, 19 mo. 5 da., 34%. 
$1,563.00, 8 mo. 20 da., 5%. 
$998.45, 87 da., 4%. 
$2,575.50, 149 da., 3%. 

10. $693.27, 214 da., 6%. 


OM WH A 


1053. Find the value of z: 

11. Principal, $240; rate, x; interest, $32.04; time, 2 yr. 11 
mo. 18 da. 

12. Principal, x; rate, 6%; amount, $717.40; time, 3 yr. 3 
mo. 4 da. 

13. Principal, $360; rate, 3% ; interest, $48.87; time, 2. 

14. Principal, $288; rate, 24% ; amount, $307.22; time, 2. 


15. Principal, z; rate, 6%; interest, $13.10; time, 4 mo. 


ll da. 


16. Principal, $270; rate, x; amount, $273.27; time, 3 mo. 


19 da. 


1054. Distinguish between “term” and “time.” Term of a 90-day 
note is 93 days. (See Arts. 939 and 941.) 


ce 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 


Term, «; face, $600; discount, $6.30; rate, 6%. | 
Term, 33 days; face, x; proceeds, $397.80; rate, 6%. 
Time, 90 days; face, $300; proceeds, x; rate, 6%. 
Term, 21 days; face, $600; discount, $2.45; rate, 2. 
Time, 4 mo.; face, $200; discount, 7; rate, 6%. 

Term, 132 days; face, x; proceeds, $2,689.50; rate, 6%. 
Term, x; face, $150; proceeds, $147.75; rate, 6%. 
Term, x; face, $1,650; discount, $4.95; rate, 6%. 

Time, 69 days; face, x; proceeds, $469,380; rate, 6%. 


LONGITUDE AND TIME. 471 


REVIEW. 
1055. Find products: 

1 648x4 11. 1,864 x 250 
2. 976x138 1208 3 9600% OL 
3. 1,648 x 874 13. 1,576 x 624 
4. 2,592 x 918 14. 176 x 232 
5. 2,416 x 875 15. 1,128 x 375 
6. 874x9¢ 16. 895 x 444 
7 848x125 17. 864 x 486 
8. 375 x 999 185/975 X313 
9. 192 xX 25 19. 372~x 64 

10. 457 x 16 20. 483 x 42 


LONGITUDE AND TIME. 


1056. Norr.— Making diagrams, as shown below, may assist the pupil 
to solve the problems. 


1. Given the longitude of A as 95° east, and that of B as 74° 
east, and the time at A as 1: 30 p.m., to find the time at B. 


Since the latitude of B has no bearing upon its time, both places may be 
located upon the same line running east and west. 


Time difference =? hours. 
Time =? Time 1: 30 p.m. 
B A 


| = Hast 
0° 74° 95° 
Longitude difference = 21°. 

Locate the prime meridian (that of 0°), then the meridians of 74° and 95° 
east. Mark above the last two the names of the places, Band A. Write 
above A its given time, 1: 30 p.m. 

To find the time at B, we must find the difference of time between B and 
A. The difference in longitude is 95° — 74° = 21°. The difference in time 
is (21 + 15) hours. 

Notr. — Remember that the more easterly of the two places has the later 
time. 


4792 ARITHMETIC. 


2. A is situated in 71° west longitude, B in 107° west longi- 
tude. What time is it at B, when it is noon at A? 
Time diff. =? 
Time? 12M. 
te] A 
West —— DEEPER EME Re STL ae 
LOT? (ey 0° 
Long. diff. =? 
3. Find the longitude of B, whose time is 8:10:80 a.m., 
when it is 7:15 a.m. at A, whose longitude is 156° 48! west. 


Time difference =? 


7:15 aM. 8:10:30 a.m. 
A B = 
West ——_—_—_ |_—_—_|_—_ East 
; 156° 487 Longitude =? 0° 


Longitude difference =? 


Since B has the later time, its location is east of A. The difference in 
time, being nearly an hour, shows the difference in longitude to be nearly 
15°. Find the exact difference. Is it to be added to 156° 48’ or subtracted 
from it, to give the longitude of B? 


4. When it is 2:40 a.m. at A, in 57° 24' west longitude, it 
is 10 a.m. at B. Find the longitude of B. 


Time difference = 71 hours. 


2:40 a.m. 10 a.m. 
A B 
West a a a East 
57° 24! cI @ hd Longitude =? 


Longitude difference = 15° x 71 = 110°. 

If we go 110° eastward from A, we shall reach the prime meridian after 
going how many degrees and minutes? How many more degrees and 
minutes must we travel to reach B? Is B in east or in west longitude? 

5. When it is noon at B, what is the time at A, the former 
being in longitude 44° east, and the latter in longitude 57° west? 


Time difference =? 


Time =? 12 m. 
A B 
West ———_| cc — East 
57° Oe 44° 


Longitude difference = 101°. Why? 


SQUARE ROOT. 473 


1057. Find the longitude or the time: 


Longitude of A. Time at A. Longitude of B. Time at B. 
6. 63° east 9 A.M. 54° east ? 
7. 57° 25! east ? 83° 20' east 1:45 p.m. 
8. 156°48! west 3:15 p.m. ? 4:10 pa. 
9. ? 11:42 a.m. 56° 25! west 1:27 P.M. 
10. 2° 15! west 6:53 A.M. 67° 48! east ? 
11. 27°10! east ? 27° 10! west 12 M. 
12. ? 4:10 P.M. 18° 4! east 11:30 a.m. 
13. 74°56! west 3:50 a.m. ? 11 a.m. 
14. 4! 30" east 8:47 aM. 90° 15’ west ? 
15. ? 10:30pm. 32°30! east 6:48 P.M. 


SQUARE ROOT. 
1058. Find the square root: 


1. os 6. rer 9. ooo? 
2. % 6. 2%, 10. 7224 
3. aT 7. S245 TL. 2304 
4. 2 8. sear 12. 2033 


Notre. — Before extracting the square root of the following, reduce tho 
mixed numbers to improper fractions. 


13. 121 16. 4124 19. 1561 
14. 112 17. 6285 20. 264, 


474 


ARITHMETIC. 


SPECIAL DRILLS. 


1060. Find sums: 


11+ 55+ 99 
46 + 22+ 88 
33 + 76 + 37 
66 + 42+ 54 


92+18+ 48 
36+ 71+ 57 
89 + 28 + 64 
53 + 47+ 35 


LOGL. Find answers: 
150 — 23 —48 162—(26+ 61) 174—41—35 165 — (28-4 47) 
172—19—66 154—(86+44) 163 —38—43 171 —(82+ 33) 
183 — 87 —42 184—(89+ 35) 155—49—24 180—(18 + 28) 
161—12—71 173—(57+17) 181—47-— 33 153 —(45+ 31) 


1062. Find products: 


36 x 31 
36 X 29 
45x 41 
45 x 39 


54 x 51 
54 x 49 
63 x 61 
63 x 59 


1063. Find quotients: 


576 +18 
693 + 21 
608 -- 19 
848 + 16 


448 +14 
533+ 13 
924 + 22 
943 + 23 


1064. Find answers: 


3815 x 144% 
3878 x 14% 
971+ 3 
85i-+ 11 


32 x 804 
45 y 1914 


1054 + 12 
109i 4 


25 + 84+ 58 
66 + 15 + 96 
27 + 19+ 87 
69 + 73 + 38 


1 2xopL 
72 x 69 
SLL 
81 x 79 


600 + 24 
675 + 75 
825 + 75 
525 + 75 


Tix Th 
81x 8} 


155i-+ 7 
2002 +18 


45 + 56 -L 68 
75 + 34 + 86 
41 + 65 + 59 
52+ 39+ 78 


90 x 89 
90 x 91 
99 x 99 
101 x 99 


225 -- 124 
150+ 61 
825 + 374 
750 + 624 


182 x 54 
124 x 64 
194 + 22 
252 + 34 


REVIEW. 475 


1065. Oral Problems. 


1. Paid 92% for coffee, 48¢ for butter, and 18¢ for lard. 
How much was my bill? 


2. I had $150. Spent $28 for a suit of clothes and $48 for 
tools. How much was left? 


3. What is the area of a field 36 yd. by 31 yd.? 
4. 600 hours = how many days? 

5. What is the cost of a cow if I pay $630 for 15? 
6. How many ounces in 294 lb. ? 


7. 1094 lb. sugar are divided among 4 people. What is the 
share of each? 


8. At 1,9,¢ per lb., how many lb. iron can I get for $5.70? 
9. What will be the cost of 51 tons iron at $17 per ton? 


10. What will be the average age of 9 boys, each 12 years old, 
and 6 boys, each 10 years old ? 


11. At 42 miles per hour, how long will it take a train to go 
882 miles? 


12. At 25¢ per hour, what will a man earn in 18 days of 10 
hours ? 


13. What will be the net price of an article whose catalogue 
price is $20.00, the discount being 90 and 10%? 


14. A man had $181 in bank. What will be his balance 
after taking out $47 and $33? 


15. How many feet in 14 rods? 
16. 77 yards = how many rods? 


17. How many sq. yd. are there in a floor 102 yd. long and 
64 yd. wide? 


18. Cost of 372 eggs at 15¢ per doz. 


476 ARITHMETIC. 


19. A man owns 3 farms containing 65 acres, 86 acres, and 
98 acres, respectively. How many acres does he own? 


20. What is the area of a piece of glass measuring 8} by 61 
inches ? 


21. What is the value in U.S. money of 50 marks at 23,8, 
cents? 


22. How many francs will a calf cost, if 18 are worth 630 
francs ? 


23. A man spends $1,740 per year. What is the average 
amount spent per month? 


PROPORTION. 


1068. A ratio is generally expressed by the sign (:). This is 
another form of the division sign (+). 


1069. Two equal ratios form a proportion. 


3+9=13 + 39; 
or, 3:9 = 13: 39; 
or, 3:9::138:39. 


1070. Supply missing term: 


1¢ 21 Laie! x 
Me Fhe) 4: 2590 
2. 834+ 16=4+2 5.538 = a= 12-420 
rR ay Ged B22 Sh 6. ?:19::28: 76 


7. 11b.1 0z.:2 lb. 4 02.::178: xf. 
8. 8 qt. 1 pt.+1 gal. =2f + 80¥. 
9. 4 bottles : x bottles = 6 pints : 15 pints. 


10. x men: 9 men = 16 acres : 36 acres. 


PROPORTION. 477 


1071. The first and the last term of a proportion constitute 
the extremes; the second and the third, the means. 
5 . 18 Sis 9 ‘ ah 
5 and 27 are the extremes. 15 and 9 are the means. The proportion is 
read: 5 is to 15 as 9 is to 27. 


1072. The proportion 3:4::2%:y may be. written ; 


x 


Clearing of fractions, we have 3y = 42; 1.., the product of the 
extremes is equal to the product of the means. 


Solve the following: 


1073. Make the product of the extremes equal to the product 
of the means, after canceling any factor common to an extreme 


and a mean. 3 

per itr Rog DAY | %=15, Ans. 

2. Sieh B80 z= 18, Ans. 

4 

3. 3: Abi: a: IB ; 

4. 3:4¢::5:2 9. 2:a::4:2 

Sepa elo a ee 10. £:8::%:24 
Gece lO 1S x RM ei ae ie? 
7 3:d4::2:44 Lo eg eee leo 
Bet ne? ease ea 13u eu Oa: 45a 


1074. Oral Problems. 
1. If 9 eggs cost 25%, what will 3 dozen cost? 
2. If 7 lb. flour cost 23%, what will be paid for 49 lb.? 
3. For $51 can get 12 straw hats. How many can I get 
for $20? ue 
4. A wheel makes 75 revolutions in 5 minutes. How many 
does it make in an hour? 


478 ARITHMETIC. 
5. $100 principal gives $6 interest. How much will be the 
interest of $450 principal ? : 


6. A merchant pays 75% freight for 125 Ib. of merchandise. 
How much will be the freight on 1,000 lb. at the same rate ? 


7. A locomotive goes 3 miles in 4 minutes. How far does it 
go in an hour? 


8. 4 horses can eat a certain quantity of hay in 10 months. 
How long will it last 20 horses ? 


9. 12 men can do a piece of work in 15 days. How long 
will 386 men require? 


10. 15 kilos cost 270 francs. What will be the cost of 5 kilos? 


1075. Slate Problems. 
NotE. — Solve by proportion or in any other way. 


1. If 9 cows cost $267, what will be the cost of 86 at the 
same rate? 


In solving such examples by proportion, we say 
9 cows = $ 267, 
36 cows = $a. 


The ratio of the cost, 267 : 2, must be the same as the ratio of the number 
of cows, 9:36. Making the proportion, we have 


9230220) 6a 
Canceling, x = $1,068. 
2. 7 bbl. sugar cost $104.32. Find the cost of 42 bbl. 


3. A wheel makes 248 revolutions in 5 minutes. How many 
does it make in 1 hour 20 minutes? 


4. A locomotive goes 1.8 kilometers in 4 minutes. How far 
does it go in an hour? 


COMMERCIAL DISCOUNT. 479 


5. From 9 kilos (kilogrammes) of yarn are made 42 meters 
of cloth. How many meters of cloth can be made from 165 kilos 
of yarn? 

How many kilos of yarn are needed for 196 meters of cloth? 


6. If 17 men receive $357 for a week’s work, how much 
should 24 men receive ? 


7. If 17 men take 27 days to finish some work, how long 
would it take 54 men? 
17 men take 27 days 
54 men take x days 
17 or 54 men?: 17 or 54 men? :: 27 days: « days. 
8. When a sum of money is divided among 48 persons, each 


receives $27.50. How much would each receive if the same sum 
were divided among 16 persons? 


9. For $85 I can purchase 238 yards of dress goods. How 
many yards can I purchase for $5? 


10. A can do a piece of work in 6 days, B can do it in 7 days. 
If B’s wages are $2.10 per day, how much should A receive per 
day?. 

COMMERCIAL DISCOUNT. 

1076. Oral. 


When the list price is $1, what is the net price after the 
deduction of each of the following discounts? 


1. 30 and 20% 6. 10 and 5% 
2. 40 and 10% 7. 20 and 20% 
3. 25 and 40% 8. 331 and 10% 
4. 50 and 10% 9. 20 and 15% 
5. 40 and 20% 10. 30 and 15% 


480 ARITHMETIC. 


1077. What single discount is equal to each of the following 
double discounts? 


11. 30 and 30% 16. 30 and 10% 
12. 20 and 25% 17. 40 and 5% 

18. 25 and 20% 18. 50 and 20% 
14. 15 and 30% 19. 40 and 15% 
15. 40 and 30% 20. 50 and 15% 


1078. Slate Exercises. 
Which is the better discount for the buyer? 


21. 30 and 20%, or 40 and 10%. 
22. 50 and 10%, or 40 and 20%. 
23. 20 and 20%, or 30 and 10%. 
24. 20 and 15%, or 830 and 5%. 
25. 30 and 15%, or 25 and 20%. 
26. 30 and 30%, or 50 and 10%. 
27. 40 and 30%, or 20 and 50%. 
28. 40 and 5%, or 30 and 15%. 
29. 20 and 50%, or 60 and 10%. 
30. 40 and 15%, or 30 and 25%. 


1079. Find the value of x: 

List price, $250; selling price, $x; rate, 40 and 10%. 

List price, $ 800; selling price, $684; rate, x and 5%. 

List price, $2; selling price, $90; rate, 334 and 10%. 

List price, $600; selling price, $378; rate, 30 and x%. 
List price, $16; selling price, $x; rate, 30 and 20%. 


SQUARE ROOT. 481 


List price, xf; selling price, 27¢; rate, 50 and 10%. 
List price, $5; selling price, $3.20; rate, x and 20%. 
List price, $x; selling price, $73.50; rate, 30 and 80%. 
List price, $200; selling price, $v; rate, 25 and 20%. 
10. List price, $1.50; selling price, 60%; rate, 50 and «%. 


OM AA 


SQUARE ROOT. 
1080. Find the square root of 425,104. 


65 2 
| 42!51'04 

19)5 651 

130|2 2604 Ans. 652. 


LOS1. In finding any figure of the root after the first, we multiply the 
other figure or figures by 2 for a trial divisor. 


1082. Find the square root of 20,857,489. 


4567 
20'85'74'89 
g5 «A 85 
mi Re 
912\7 63889 Ans. 4,567. 


1083. Find the square root of 


1. 64,516 6. .702244 
2. 734.41 7. 264.7129 
3. 1.8769 8. .23775376 
4. 718.24 9. .093636 
5. 14.1876 10. .004761 


489, ARITHMETIC. 


1084. Slate Problems. 


1. What is the profit on 9 boxes of oranges, each containing 
20 dozen, bought at $1.10 per hundred and sold at the rate of 
18 for 25¢? 


2. How long will it take a train to go 176 miles at the rate 
of 3,520 feet per minute? 


3. If .0875 of an acre of land is worth $9, what is 3, acre 
worth? | 


4. At £1 1s. 7d. per barrel, how many barrels of flour can 
be bought for £161 17s. 6d. ? 


5. A,B, and C buy a house for $7,500. A furnishes $2,000; 
B, $2,500; C, the remainder. The yearly rent, less expenses, is 
$576. To what amount is each entitled ? 


6. If 580 tiles, each 6 inches square, will cover a certain 
area, how many tiles, each 4 inches long and 3 inches wide, 
will be needed to cover the same area? 


7. A man receives $1,500 commission on his yearly sales. 
What is the amount of his sales, if he is allowed + per cent com- 
mission ? 


8. At what rate per cent will $360 produce $3.06 interest 
in 2 months 12 days? 


9. Find the square root of 25.00400016. 


10. What will be the capacity, in gallons, of a tank 9 feet 
long, 6 feet 8 inches wide, and 6 feet 5 inches deep? 


11. What decimal multiplied by 312.5 will give the sum of 8, 
zs, %, 09375, and 2.46? 


12. A dealer bought a lot of coal at $4.95 per ton. What 
was the total cost if he gained $142.50 by selling it at $5.25 per 
ton? 


FRACTIONS. 488 
QL + 4.5, 
18x38 


14. The front wheel of a wagon measures 18 feet in circum- 
ference. What is the distance traveled in miles, rods, yards, etc., 
when the wheel has made 527 revolutions? 


15. Write in words .349, 300.049, 34%, 300549. 


13. Find the value of —4 of 63. 


16. If a bar of silver weighing 4 lb. 6 oz. 12 pwt. is worth 
£13 8s. 4d., what is the value (an English money) of a similar 
bar weighing 7 lb. 9 oz. 12 pwt.? 


17. A and B form a partnership. A furnishes $5,000; B, 
$10,000. During the year A draws $1,500 of the profits and 
B draws $1,000. At the end of the year the entire business is 
disposed of for $20,000. What amount should each receive ? 


18. What per cent is gained on en article bought for 20 per 
cent less than its value and sold for 20 per cent more than its 
value? 


19. A person loans $750 to M and $1,200 to N at the same 
rate. From the latter he receives half-yearly $9 more interest 
than from the former. What is the annual rate of interest ? 


20. A 4-months note for $375, drawn March 19, was dis- 
counted at a bank June 4. Find the proceeds. Rate, 6%. 


21. M can do a piece of work in 4 days, N can do it in 5 
days, 0 in 6 days. How long will it take the three together to 
do the work? 


REVIEW FRACTIONS. 
1085. Slate Exercises. 


1. Divide the sum of 63 and 1% by the difference between 
24 and 34, 

2. What is the difference between the sum of # and 3 and 
the product of $ and 54? 


A84 ARITHMETIC. 


3. What is the product of the sum and the difference of 44 
and 64? 


4. Subtract 2 of + from 44; and find the value of 5% of 
16s. 6d. 


5. Add 78, 8 of zy, & of 72, and 43. 


6. Reduce $ of asq. rod to the fraction of an acre, and find 
the value of ;& of a ton in pounds and ounces. 
34 — 24 
34+ 2h 


7. Reduce 58&%6& to its lowest terms, and to its 


simplest form. 


8. Addi, 3, #,and4; multiply the sum by %; and sub- 
tract the product from 1. 


9. Find the value of 954 meters at 43 francs per meter. 

10. Divide 21 by 34, and add the quotient to 55. 

11. Multiply 2,2, by 163, and divide the result by 14 of 22. 

12. Reduce 7s. 6d. to the fraction of a £, and 7hr. 12 min. to 
the fraction of a day. 
2+ 4 of 54 

Foy 
14. Add together £2 and + of 53 shillings. 
15. What fractional part of 7 A. 127 sq. rd. is 5 A. 81 sq. rd.? 
16. What must be added to $ of to make it equal to 7% of 32? 
17. 2 of a number is 148. What is the number? 
18. If? of a field is worth $325, what is the field worth? 
19. If 2 of a house is worth $4,900, what is the value of 4? 
20. If 3; of a ship is worth £1,278 2s. 6d., what is 5; worth ? 


fs = ay = £1,273 2s. 6d. 
Deduct = 


13. Reduce to its simplest form 


SQUARE ROOT. , 


DOMESTIC EXCHANGE. 


1087. Slate Exercises. 


Find the value of x. 


485 


(Do not allow days of grace in the case of “sight” drafts.) 


FACE oF 


DRAFT. 


x 
. $1,800 
x 
$200 
$ 600 
. $1,000 
. $1,200 
$800 
$400 


RATE oF EXCHANGE. 


$100 20% per M. premium 


par 
x 


1% premium 


$1 per M. discount 


38% premium 


75¢ per M. discount 


par 


x 


$2 per M. premium 


1088. Slate Problems. 


TERM. 
sight 
6 days 
60 days 
sight 
30 days 
Dp 
z 
93 days 
24 days 
9 days 


Rate oF Cost oF 


IntEREST. DRAFT. 
x 
6% $499.50 
6% $1,778.85 
$701.75 
6% x 
6% $598.95 
6%» $999.25 
z $1,178.30 
6% $796.80 
az $400.80 


APPLICATIONS OF SQUARE ROOT. 


1. How many inches in the side of a square table top con- 
taining 529 square inches? 


97 sq. in. 


(Reduce area to square inches.) 


2. The surface of a square piece of board contains 3 sq. ft. 
What is the length of one side in feet and inches? 


3. How many rods long is a square field containing 90 acres? 
How many yards of fence would be needed to enclose it? 


486 _ ARITHMETIC. 


4. Land surveyors use a measure called a chain. What is 
its length in feet, 10 square chains being equal to an acre? 
It is subdivided into 100 “links.” Find the length of a link 


in inches and decimal. 
» 


5. The surface of the six equal faces of a cube is 1,350 sq. 
inches. What is the length of the cube? 


6. Carefully construct a right-angled triangle, base 4 inches, 
perpendicular 3 inches. Measure the hypotenuse. 
Take the square of the length of each side, and endeavor to 
show the relation between the square of the hypotenuse and the 
squares of the other two sides. 


7. Construct a right-angled triangle, base 3 in., perpendic- 
ular 14 in. Measure the hypotenuse, and see if the relation 
between this hypotenuse and the other two sides of this triangle 
is the same as that found in the other triangle. 

8. A right-angled triangle has a base 12 inches long; its 
perpendicular is 34 inches. What is the length of the hypot- 
enuse ? 


9. The hypotenuse of a right-angled triangle is 25 inches; 
its perpendicular is 7 inches. What is the base? 


10. The base of a right-angled triangle is 12 feet; the hypot- 
enuse is 13 ft. Find the perpendicular. 


1089. Draw a right-angled triangle (Fig. 1). Upon each side 
construct a square (Fig. 2). From the upper portion of the 


lpg 


Fira. 1. Fia. 2. Fia. 3. Fia. 4. Fia. 5. 


COMPOUND INTEREST. A487 


largest square C, cut a right-angle triangle of the same dimen- 
sions as those of the original triangle m. Cut another triangle 
of the same dimensions from the left-hand portion (Fig. 3). 
Place one of these triangles below the remainder of the square C, 
and the other at the right, as in Fig. 4, and the resulting polygon 
will be exactly equal in surface to the two squares A and B 


(Fig. 5). 
COMPOUND INTEREST. 


1091. Slate Exercises. 
1. Find compound interest of $2,048, for 3 years, at 5%, 


interest compounded semi-annually. 


$ 2,048 Do not use unnecessary figures. 75 each 6 mo. 
first 4 year 51.20 
$ 2,099.20 
second year 52.48 Carry to only 4 places of decimals. 
$ 2,151.68 
etc. etc. 


2. Compound interest of $1,864, at 4%, for 2 years, interest 
compounded quarterly. 


3. Compound interest of $1,500, at 6%, for 3 years, interest 
compounded semi-annually. 


4. Amount of $800 at compound interest, for 3 years, at 3%, 
interest compounded semi-annually. 


$ 800.00 

19 8.00 

first 4 year { 1 4 4.00 

$812.00 

0, 

second 4 year { Fe ie 

$ 824.18 
4 19 8.2418 
third 4 year { i 4.1209 
$ 836.5427 


etc. etc. 


488 ARITHMETIC, 


STOCKS AND BONDS. 


1092. Slate Problems. 


Brokerage is calculated on the par value. The dividends are based on 
the par value. 

1. Find the cost of 240 shares mining stock, par value $10, 
at 873, brokerage 4%. 


2. Paid $11,460 for 120 shares R. R. stock, par value $100, 
brokerage 4%. What was the value of the stock per share? 


3. Bought 150 shares canal stock at 874, brokerage 1%, pay- 
ing for it $5,265. What is the par value per share? 


4. How much brokerage is paid by the buyer of 275 sharer 
bank stock, par value $100, brokerage 1% ? 


5. A broker sells for a customer 200 shares stock, par value 
$25, at 1024. If he retains 1% brokerage, how much does he 
pay over to the former owner of the stock ? 


6. A man buys 60 shares bank stock, par value $100, at 40, 
no brokerage. If the annual dividend is 18%, what is his income 
therefrom ? What per cent does he receive on his investment ? 


7. A manufacturing corporation makes $20,000 a year over 
all expenses. The stock consists of 4,000 shares, par value $50. 
What rate of dividend can be declared ? 

What per cent on his investment does a man receive who has 
bought his stock at 175, no brokerage? 


8. A railroad company’s stock consists of 10,000 shares, par 
value $100. Its profits for the year are $47,500, out of which 
must be paid the interest for the year on $200,000 worth of 
bonds, at 5%. What rate of dividend can be paid the stock- 
holders ? 


9. A capitalist bought 360 shares stock, par value $25, at 
1683. He paid therefor, including brokerage. $15,176.25. 
What was the rate of brokerage? 


SQUARE ROOT. 489 


10. A broker sold 250 shares, par value $100, at 107%. He 
deducted brokerage, and paid over the proceeds, amounting to 
$26,875. Find the amount of the brokerage and the rate per 


cent. 


11. A woman invests $35,050 in stock at 175, brokerage 1%. 
If the annual dividends are 74%, what is her income from the 
investment ? 


12. Mr. Tower pays $104 for a $ 100 five per cent bond. At 
the end of six years, the bond is redeemed at par. What rate of 
simple interest does he receive on his investment of $ 104? 


SQUARE ROOT. 
1094. Slate Exercises. 


Find square roots to three decimal places: | 
he DOL) 3. 38 5. 350 (by aye 9. 1,874 
2. 14 4. 74 6. 758 8. 1,384 10. 4,000 


1095. Oral Exercises. 
What is the square of 1? Of.38? Of.11? Of LZ? 


1096. How many decimal places in the first two answers? 
In the last two? 


V/01 =? V.09 =? 0121 =? V.0144 =? 


1097. Slate Exercises. 


Find square roots to three decimal places: 


1.0 Ger 4 9. 3.6 Lae Lee lak 
2. 40 6. 9 10. 1.60 14. 64 18. .144 
saa Taal 11. 2.50 LGsoek 193169 
4341 8. 2.5 12. 3.60 16. 10.0 20.000 


490 ARITHMETIC. 


MEASUREMENTS. 


1099. Slate Exercises, 


Find the missing side of each of the following ten right-angled 
triangles : 


1. Base, 15; perpendicular, 8; hypotenuse, 2. 
2. Base, 35; perpendicular, x; hypotenuse, 37. 
3. Base, 2; perpendicular, 15; hypotenuse, 39. 
4. Base, 20; perpendicular, 21; hypotenuse, 2. 
5. Base, x; perpendicular, 45; hypotenuse, 53. 
6. Base, 56; perpendicular, 2; hypotenuse, 65. 
7. Base, 55; perpendicular, 48; hypotenuse, 2. 
8. Base, x; perpendicular, 14; hypotenuse, 50. 
9. Base, 63; perpendicular, x; hypotenuse, 65. 
10. Base, 112; perpendicular, 15; hypotenuse, 2. 


1100. Slate Problems. 


11. One parallel side of a trapezoid measures 160 yd., the 
other measures 200 yd., the area is 32,400 sq. yd. Find the 
perpendicular. 


12. One parallel side of a trapezoid is 20 rods, the perpendic- 
ular is 15 rods, the area is 225 sq. rods. Find the length of the 
other parallel side. 


13. One parallel side of a trapezoid measures x rods, the other 
measures x + 6 rods, the perpendicular is 10 rods, the area is 150 
sq. rods. Find the length of the parallel sides. 


14. Find the area in acres of a right-angled triangle, the 
length of the sides being 24 rods, 7 rods, 25 rods. 


REVIEW. 3 49] 


15. A court yard 84 ft. by 36 ft. is to be paved with flag- 
stones measuring 6 ft. by 3 ft. How many stones will be needed ? 
What will be the cost of the work at $1.25 per sq. yd.? 


pane 


1 chain = 66 feet. | 1eicHaine. 
Norr. — A right angle contains 90 degrees. 


16. How many rods of fence will be 
needed to enclose the field shown in the 
diagram ? 


12 chains, 


70 rd. 
17. Find the length of the fourth sideof [ys 
the following piece of ground. E 
How many yards of fence are needed to yh ranr 


enclose it? How many acres does it contain? 


18. What is the length of the diagonal of a rectangular field 
90 yd. wide, 120 yd. long? 


19. The dotted line in the accompanying 
diagram indicates a path through the field. 
How many yards are saved by taking the 
path instead of following the road? Sandi 


86 chains. 


27 chains. 
Road. 


20. Find the length (in rods and a decimal) of the diagonal 
of a square 40-acre field. 


REVIEW. 
1101. Oral Problems. 
To the following ten problems the wrong answers are very frequently 
given. 

1. Sold a horse for $250, losing $50. What is the loss per 
cent ? 

2. If 3 boys solve 3 problems in 3 minutes, how long will it 
take 6 boys to solve 6 problems? . 

3. Two boys go fishing; one brings 7 cakes for lunch, the 
other brings 5 cakes. A third boy joins them at noon, and pays 
12¢ for his share of the dinner. How should the first two 
divide the money received ? 


492 ARITHMETIC. 


4. If 100 per cent is gained by selling an article for $1, how 
much would be gained by selling it for $2? 


5. A boy had a slate 5 inches by 7 inches. He buys one 
twice as large. Give the dimensions of the new slate. 


6. A man wishes to put up on the front of his lot a fence 30 
feet long. If the posts are 6 feet apart, how much will they 
cost at 25 ¥ each? 


7. One-half the money received by a newsboy is profit. 
What per cent does he make? 

8. 50 per cent of a number multiplied by 30 per cent of the 
same number equals 60. What is the number? 

9. Three-fourths per cent of a number is 90. What is the 


number ? 


10. An importer receives some cases of goods numbered con- 
secutively. How many cases are there, if the number of the first 
is 28, and of the last 75? 


1102. Slate Problems. 


1. If a bar of silver weighing 2 lb. 3 oz. 6 pwt. is worth 
£6 18s. Td., what is the value in English money of a oe bar 
weighing 15 lb. 7 oz. 4 pwt.? 


2. A quantity of provisions would last a ship’s crew 20 days, 
allowing each man 2 lb. 4 oz. daily. What should each man be 
allowed so as to make the provisions last 4 days longer? 


3. If 40 men are able to do a piece of work in 10 hours, how 
many extra men must be employed to finish it in 8 hours? 


4. Ifit requires 40 yd. carpet 2 ft. 9 in. wide to cover a floor, 
how many yards of carpet 2 ft. 6 in. wide would be needed ? 


5. How long will it take a train to go 112 miles, at the rate 
of 46 miles in 1 hour 20 min. 30 sec. ? 


6. Change 1,759 yards to rods, yards, ete. 


REVIEW. 493 


7. If a beam 5 ft. 6 in. long, 10 in. wide, and 8 in. thick, 
weighs 924 lb., find the length of another beam of the same 
material which weighs 3,024 lb., and whose end is a square foot. 


8. A field 110 yd. long and 44 yd. wide contains an acre. 
What is the area of a field 220 yd. long and 88 yd. wide? Of 
one 440 yd. long and 176 yd. wide? 


9. A ship with a crew of 32 men has provisions that will 
serve for 45 days, at a daily allowance of 3 lb. for each man. If 
it then picks up another vessel's crew consisting of 16 men, what 
must be the daily allowance, to make the provisions last for 40 
days? } 


10. If a steel bar 12 ft. long, 4 in. broad, and 24 in. thick, 
weighs 480 lb., what is the weight of another steel bar 18 ft. 
long, 3 in. broad, and 2 in. thick ? 


11. If 8 horses eat 13.5 bushels of oats in 9 days, how many 
days will 15.75 bushels last 14 horses? 


12. A person deposits in two banks $750 and $1,200, respec- 
tively, at thesamerate. The latter sum draws $18 more interest 
per year than the former. What is the rate per cent? 


13. Two men have saved $2,000 each. One has loaned 
$1,400, at 4%, and the remainder at 5%. What rate must the 
other man receive for his money in order to get the same 
interest ? 


14. I owe $8,625, payable in 3 years 4 months. I have at 
present $7,500. What rate of interest must I receive to pay my 
debt at maturity ? 

15. A person loans 4 of his capital at 5%, and the other half 
at 4%. He receives annually $40 more interest from the former 
than from the latter. What is his capital? 

16. A certain sum loaned at 4% produces $30 less interest 
than a sum $400 greater, loaned at 5%. How muoh is loaned 
at each rate? 


494 ARITHMETIC. 


17. A capitalist has placed 4 of his money at 4%, and his 
remainder at 5%. His income is $2,940 per year. What is his 
capital ? 


18. Change 13,576 inches to rods, ete. 


19. Three men buy a lot for $600. After selling it A receives 
$ 220 as his share of the proceeds, B receives $280, and C $3800. 
How much did each invest originally ? 


20. D receives 4 of a sum of money, E 4, and F the remain- 


der. E’s share is $90 more than D’s. What is the share of F? 


21. A man receives $593.70 as the proceeds of his note. 638 
days thereafter he pays the bank $600. What rate of interest 
has the bank charged on the $593.70 loaned ? 

What is the rate of bank discount on the $600 note? 


22. A tank is fed by two pipes, one of which can fill it in 2 
hours, and the other in 3 hours. A third pipe can empty it in 
1 hour. If, when the tank is full, the supply pipes and the 
exhaust pipe are all set to work, in what time will it be emptied ? 


23. What per cent is gained on oranges bought at 20 cents 
per dozen and sold at the rate of 10 for 25 cents? 


EXACT INTEREST. 


Exact interest is used by the Government in its calculations. 365 days 
are taken to the year. 


1104. Slate Exercises, 


1. Find the exact interest of $280 from April 14 to Sept. 6 


at 4%. 


Time 145 days. Ans. $280 x —*- x 14. 


100 365 


2. Find the exact interest on $76.65 from March 4 to Dee. 
15 at 6 per cent. 


MEASUREMENTS. 495 


On $384 at 733 per cent for 75 days. 

On $ 438 at 5% from Jan. 1 to March 15. 

On $ 109.50 at 42% for 87 days. 

On $847.60 at 5% from April 29 to Sept. 20. 
7. On $584 at 32% from May 16 to Dee. 1. 


Oo on FP & 


1105. Unless “exact” or “accurate” interest is specified, use 360 days 
to the year. 


MEASUREMENTS. 


1. What is the area of a triangle whose sides measure 15, 16, 
and 17 inches, respectively ? 


15 From the half sum of the three sides subtract each 

16 side separately. The square root of the continued prod- 

iA uct of the half sum and the three remainders will be the 
2)48 area. 

ea V2EXIXBEXT= 

24—16=8 


04—17=7 12,096 = 109.98 sq. in. Ans. 
2. Find the area in square feet of a triangle whose sides 
measure 386 ft., 84 ft., 91 ft. 


3. Find the area of a triangle whose sides measure 21, 28, 
and 35 rods, respectively. B 


4. In the following field, AB measures 
59 rods; BC 52 rods; CD, 25 rods; AD, 
60 rods; and the diagonal, AC, 65 rods. 4 C 
Find the area of the field in square rods. 


5. Find the area of an isosceles tri- D 


angle whose base is 30 yards, its equal sides measuring 25 
yaras. : 

6. What is the altitude of an isosceles triangle, base 96 ft., 
equal sides 64 ft.? Find its area. 


4.96 ARITHMETIC. 


7. Find the area of an equilateral triangle, each side being 
6 ft. 

8. Find the area of a right-angled triangle, base 42 ft., 
hypotenuse 70 ft. 

9. Find the area of an isosceles triangle, altitude 48 ft., equal 
sides 50 ft. 


10. Place two equilateral triangles, sides 2 inches, base to base, 
making a rhombus. Find its area, also the length of each di- 
agonal. 


11. Find the radius of a circle whose circumference is 1382 ft. 
(3.1416 x diam. = circum.) 

12. Find the area of a circle whose radius is 4 inches. (Area 
= circumference X 4 diameter.) 

13. Find the area of a circle whose radius is x inches. 

14. Find the radius of a circle whose area is 314.16 sq. in. 


15. Find the area of a circle whose circumference is 15.708 ft. 


PARTIAL PAYMENTS. 


1108. Merchants’ Rule, 
Brooktyy, N.Y., June 19, 1894. 
On demand, I promise to pay William R. Budd, or order, Two 
Thousand Four Hundred Fifty-four 343, Dollars, value received, 
with interest at 6 per cent. 
$2,454,705. ARTHUR TOWNSEND. 


The following payments are endorsed on the note: 


July 5, 1894, $200. 
July 29, 1894, $450. 
Sept. 18, 1894, $700. 
Oct. 25, 1894, $300. 


Find the amount due Jan. 2, 1895. 


PARTIAL PAYMENTS. 497 


If no payments had been made, there would be due $ 2,454.75 
And interest from June 19 to Jan. 2, 197 days, 80.60~ 
Total due, $ 2,535.35 
The credits are: Payment July 5, 1894, 200.00 
Interest on $200, July 5 to Jan. 2, 181 days, 6.03 
Payment. July 29, 1894, 450.00 
Interest on $450, July 29 to Jan. 2, 157 days, 178 
Payment Sept. 18, 1894, 700.00 
Interest on $700, Sept. 18 to Jan. 2, 106 days, 12.37 
Payment Oct. 25, 1894, 300.00 
Interest on $300, Oct. 25 to Jan. 2, 69 days, 3.45 
Balance due, $851.72 


1109. By the merchants’ rule, interest is calculated on the 
face of an interest-bearing note from its date until settlement, 
and interest is allowed on all credits from their payment until 
settlement. 


1110. Slate Exercises. 


1. A note for $500, with interest at 6%, is dated July 25, 
1893. Payments are made: $100, Sept. 18; $200, Feb. 5, 1894. 
How much is due April 1, 1894? 


2. Find amount due Sept. 15, 1894, on a demand note for 
$1,875, with interest at 6%, dated Jan. 18, 1894. Payments of 
$1,000 and $500 were made March 30 and June 17, respectively. 


3. June 12, 1892, Robert Colgate bought goods amounting to 
$600. Dec. 31, 1892, he paid $300; April 5, 1893, $200; June 
1, 1893, he settled the account. How much did he pay on that 
date, if he is charged 6% on the purchase from its date, and is 
allowed 6% interest on his payments? 


4. T. J. Minturn loaned Chas. A. Dorsey $500, Sept. 1, at 
6%. Payments of $200 each were made Oct. 1 and Nov. 1. 
How much is due Dec. 1? 


498 ARITHMETIC, 


Dr. Witson T. Jones. Cr. 
1893. 1893. 
Feb.| 5/To merchandise, | 840]00]Mar.| 9] By cash, 500 | 00 


Dec. | 31 | To interest to date, Sept. | 13 | By cash, 200 | 00 
SEY Bye cashe 


Dec. | 31] By interest to date, 


5. Find the amount paid in settlement of the foregoing 
account, Dec. 31, 1893. Interest 6%. 


6. A merchant’s books show the following debits: Feb. 13, 
merchandise, $725.00; April 14, merchandise, $603.00. The 
credits are: April 5, cash, $600; Aug. 29, cash, $300. How 
much is due Oct. 5, interest 6%? 


L111. The merchants’ rule is frequently used where the transactions all 
take place within a year. The exact number of days is taken, and the 
interest is calculated on the basis of 360 days to the year. 


CHAPTER XIV. 


EQUATION OF PAYMENTS. — MENSURATION OF SURFACES 
AND VOLUMES. — BOARD MEASURE. — ANNUAL INTER- 
EST. —GOVERNMENT LANDS. — METRIC SYSTEM, 


EQUATION OF PAYMENTS. 


1114. Oral Problems. 


1. A friend loans me $800 for 6 months without interest. 
How long ought I to loan him $400 to cancel the obligation? 


2. In what time would the interest on $450 be the same as 
the interest for 8 months on $600? 


3. W borrows from X $200 for 5 months and $400 for 2 
months. How much money should W loan X for one month in 
return for the accommodation ? 


4. A man offers a lot for $600, payable $300 in 2 months, 
and $300 in 4 months. How much credit should be given toa 
buyer who wishes to pay the $600 at one time? 


5. Mr. Jones has bought $600 worth of goods on 6 months’ 
credit, and $300 worth on 8 months’ credit. For what time 
should he give a note (without days of grace) for the whole 
amount, $900? 


1115. Slate Problems, 


1. In what time would the interest on $1,000 be the same as 
the total interest on the following amounts: 
$100 for 1 month, $200 for 2 months, $300 for 3 months, $400 
for 4 months? 
499 


500 ARITHMETIC. 


Interest on $100 for 1 month = Interest on $100 for 1 month 


4 “200 “ 2 months = We + SOW a ae 
“ “ 300 “c 3 a3 — « 4c 900 “ce il oc 
« « 400 « 4 a aa “cc « 1600 “ 1 “cc 
ec 46 1000 “cc 2 66 = (73 “e 3000 cc i “cc 


2. A person owes $400 payable in 4 months, and $500 pay- 
able in 18 months. What would be the average time for the 
payment of the whole indebtedness of $900? 

The debtor is entitled to the use of $400 for 4 months, which is equal to 
the use of $1,600 for 1 month. He is also entitled to the use of $500 for 13 
months, which is the same as $6,500 for 1 month. He is entitled, in all, to 


the use of $8,100 for 1 month, which is equal to the use of $900 for how 
many months? 


L116. By equation of payments is meant a method of ascer- 
taining at what time several debts due at different times may be 
settled by a single payment. The time thus found is called the 
average time, or the equated time. 


3. Find the average, or equated, time for the payment of the 


following : 
$600 due in 2 years 


$500 due in 14 years 
$300 due inl year 
$400 due in 9 months 


4, $250 due in8 months 
$450 due in 6 months 
$500 due in 8 months 
$600 payable in cash 


250 x 8 = 
450 x 6 = 
500 x 3 = 
600 x 0=0 


1800 x ? = 


EQUATION OF PAYMENTS. 501 


5. $ 200 due in 15 days 
$300 due in 30 days 
$A00 due in 45 days 


6. $840 to be paid in four equal installments in 1, 2, 3, and 4 
months, respectively. 


7. $960 to be paid + in 2 months, 4 in 4 months, 4 in 5 
months, and the remainder in 6 months. 


8. A debt to be paid ;4; in 2 months, + in 8 months, 4 in 4 
months, and the balance in 12 months. 


9. $6,000; 4 to be paid in cash, + of the remainder in 3 
months, another fourth in 6 months, and the balance in 9 
months. 


10. On what date should the following account be paid in 
full? 

Bought, July 1, goods to the amount of $300 payable in cash, 
to the amount of $800 payable in 30 days, and to the amount 
of $1,000 payable in 60 days. 


1117. Miscellaneous. 


11. A farmer sold 300 bu. wheat at 921 per bushel, 100 bu. 
at 90, 400 bu. at 95, 200 bu. at $1. What was the average 
price ? 


12. Three men hire a pasture for $84. One puts in 15 cows 
for 12 weeks, the second puts in 20 cows for 6 weeks, the third 
puts in 18 cows for 10 weeks. What amount should each pay? 


13. A and B form a partnership. A furnishes $2,000, B 
$3,000. After a year A furnishes an additional $1,000. At 
the end of 2 years the business is disposed of for $7,100. How 
much should each receive? 

SuecEstion: A receives his $3,000 and how much of the profits? 


Should he receive as much as B, who had $3,000 in the business the 
whole time? 


502 ARITHMETIC, 


14. How many bushels of bran worth 40 cents per bushel 
should be mixed with bran worth 30 cents per bushel to make 
100 bushels worth 36 cents a bushel ? 


x = number of bushels at 40 ¢ 
100 — x = number of bushels at 309 
40 « = value (in cents) of one kind 
30(100 — x) = value of other kind 


Total value = how many cents? 


15. How many bushels of corn worth 60% per bushel should 
be mixed with 80 bushels of corn worth 50 per bushel to make 
a mixture worth 52¢ per bushel? 


16. A can doa piece of work in 20 days, B can do it in 80 
days. They work together and receive $5 per day as the wages 
of both. What should be A’s share of the total amount received? 


How long does it take both together to do the work? What would A 
receive per day if he did the work alone? 


17. A partnership is formed between A with a capital of 
$1,500 and B with a capital of $2,500. Six months thereafter, 
they take in C with a capital of $4,000. How should a profit of 
$3,500 be divided at the end of the year? 


18. Three merchants shipped a cargo of iron by sea. A sent 
180 tons, B sent 105 tons, C sent 315 tons. During a storm the 
sailors were obliged to throw overboard 180 tons to save the 
vessel. What portion of the loss should each merchant sustain ? 


19. If pure milk is reduced in value from 24 per gallon to 
20¢ per gallon by the addition of water, how many quarts of 
water have been placed in a can that contains 40 quarts of the 
adulterated article ? 


20. Find the entire surface of a cube whose edge measures 7 
inches. 


21. What is the edge of a cube whose entire surface contains 
726 square inches? 


SURFACES. 503 


MENSURATION OF PLANE SURFACES. 


1124. Slate Exercises. 


1. Find the circumference of a circle whose radius is 2. 


(Diameter x 3.1416.) 


2. Find the area of a circle whose radius is 2. 


(4 circumference x 4 diameter.) 
3. Find the area of a circle whose diameter is 2. 
4. Find the area of a circle whose circumference is x. 
5. What is the area of a circle whose diameter is 86 feet ? 


6. What is the radius of a circle whose area is 158.9384 
sq. yd.? 


7. What-is the circumference of a circle whose area is 198.95 
sq. rods? 


8. Find the area of a square whose diagonal is z. 
9. Find the area of a square whose diagonal is 150 rods. 


10. Find the area of an isosceles triangle, its base being 56 
meters, equal sides 100 meters. 


11. Find the area of an equilateral triangle whose side is 12 
feet. 


12. Find the area of a triangle whose sides are 50 yd., 60 yd., 
70 yd. 


13. What is the area of a circle whose cir- 40 rd. 
cumference is 10 feet? 


(The square of the circumference x what = area?) 


14. Find the area of the rhomboid, Fig. 1. Fig. 1. 


504 ARITHMETIC. 


15. Of the rectangle, Fig. 2. 16. Of the rhombus, Fig. 3. 
17. Of the trapezoid, Fig. 4. 


25 rd. 


Fie. 2. | Fie. 3. Fia. 4. 
18. Of the trapezium, Fig. 5. 19. Of the rhombus, Fig. 6. 


20. Find the altitude, AB, of the following triangle, Fig. 7: 
(First find the area.) 


30 yd. 
A 
xy < 
4 og 2 x, 
|: § % : 
4 
5a 
24 rd. 7 ft. 
Fia. 5, Fie. 6. Viex7; 


21. Find the diagonal (in rods) of the square whose area is 5 
acres. 


22. Find the area of a hexagon, composed of six equilateral 
triangles, each side being 6 inches. Fig. 8. 


Fig. 9. Fia. 10. 


23. What is the area of the circle circumscribed about the 
above hexagon, Fig. 9? 


24. What is the area of the square inscribed in a circle whose 
diameter is 10 feet, Fig. 10? 


REVIEW. 


SPECIAL DRILLS. 


1129. Give sums: 


112+91+485 
129+ 62+ 98 
182+ 138+67 
114+ 21+49 


43+131+61 
26+172+81 
75+ 193+ 23 
1382+494-+77 


1130. Give answers: 


150 —23-+-48 
154— 36+ 44 
155—49-+ 24 
1538 —45+31 


172+19—66 
184+ 39—385 
181+47—33 
151+ 46— 24 


1131. Give products: 


44 x 20 
44 x 22 
44x18 
63 x 28 


63 x 82 
54 x 38 
54 x 42 
88 x 48 


1132. Give quotients: 


676+ 13 
602 + 14 
690+ 15 
672 +16 


527 +17 
738 + 18 
950 + 19 
924 + 21 


1133. Give results: 


84 x 13 
48 x 27 
36 x 94 
48 x 192 


2111 13 
2142 + 14 
1304 + 21 
1554 + 22 


95+ 144+ 79 
63-+117+97 
91+126+4382 
63+ 143+ 24 


183 —(72—87) 
LTS (5 nes) 
165—(47 —28) 
182—(48—38) 


83 X 52 
26 x 58 
26 x 62 
Lax OS 


704 + 22 
966 — 23 
768 + 24 
975 + 25 


36 x 498 
32 X 59f 
49 x 49 

_ 58 X 58 


505 


68+-56-+174 
63+ 34+ 186 
91+59+165 
78+39-+183 


161-4+79—12 
174+41—36 
E7i-f ooh ode 
175-433 —46 


Uti bP 
TL Gio 
TL C82 
45 x 88 


887 + 27 
961 + 31 
992 + 32 
759 + 33 


1624 + 25 
1587 + 31 
182 + 32 
173 + 5 


506 ARITHMETIC. . 


1134. Oral Problems. 


1. A has 96 sheep; B has 28 sheep more than A. How 
many sheep have both? 


2. There are 56 pupils in one class, 48 in a second class, and 
52 in a third class.) How many pupils are there in the three 
classes ? 


3. March 29 is what day of the year 1894? 


4. How far is a man from his starting-point, if he travels 
due east 150 miles, due west 23 miles, due east again 48 miles? 


5. A body falls 16 ft. in the first second, three times as far 
in the second second, five times as far in the third second. How 
far does it fall in three seconds ? 


6. The base of a right-angled triangle is 12 ft., the perpen- 
dicular is 16 ft. What is the hypotenuse? 


7. At $35 per month, what will be the rent of a house for 
16 months? 


8. A field containing 169 square rods is 18 rods long. How 
many rods of fence will be needed to enclose it? 


9. 25 packages of sugar weigh together 874 lb. How many 
pounds are there in each? 


10. At 45 miles per hour, how many hours, minutes, and 
seconds will it take a train to go 230 miles? 


11. How many years have elapsed since the invention of gun- 
powder, 1356? 


12. What profit is made on an article bought for $175, less 
12%, and sold for $200? 


13. How many square rods in a field 71 rods long, 81 rods 
wide? 


14. Assuming a kilo to be 24.1b., how many kilos will be equal 
to 143 lb.? 


MEASUREMENTS. 507 


15. A degree of longitude in latitude 45° is about 70% of the 
length of a degree on the equator. Calling the latter length 69 
miles, how long is a degree of longitude in latitude 45°? 


16. At $44 per acre, how much land can be bought for $ 968? 


17. A number of marbles divided among 29 boys gives each 
16 marbles, and leaves a remainder of 26. How many marbles 
are there? 


18. What is the monthly salary of a clerk who receives $1,500 
per year? 


19. How many revolutions in a mile, 5,280 ft., are made by a 
locomotive wheel 16 ft. in circumference? 


20. How many feet of fence are there around a lot 49 ft. 
wide, 87 ft. long? 


21. How many bricks 8 in. by 4 in. by 2 in. would make a 
cubic foot ? 


22. 13 is one factor of 1,001. Find the other two prime 
factors. 


23. What are the three equal factors of 348? 
24. What is the square root of 1,225? 


25. At 44 miles per hour, how long will it take a man to 
walk 374 miles? 


26. What will be the cost of 9 dozen hats at $1.331 each? 


MEASUREMENTS. 
1137. Find the area of each of the following triangles and its 
altitude. 


When the area of a triangle is known and the length of the base, how 
can its altitude be calculated ? 


508 ARITHMETIC. 


Base, 51 ft.; other sides, 20 ft. and 37 ft. 

Base, 21 yd.; other sides, 13 yd. and 20 yd. 

Base, 148 rods; other sides, 39 rods and 118 rods. 
Base, 28 chains; other sides, 17 chains and 25 chains. 


Base, 75 inches; other sides, 20 inches and 65 inches. 


Ce So 5. = Ne 


1138. Find the areas of the following quadrilaterals: 

6. Given AB, 17; BC, 10; CD, 20; DA, 18. AC= 21. 

7. Given 4B, 25; BC 89;.CD, 34) DA, 50s ACG: 

8. Given AB) 37; "BC, 1b CD ao DA Lia aaa 

9. Given 48,111 ;) BC 45.°CD) 2b) (AT GA Clee 

10: Given’ A 5,113; BON CD60) DAP TI aC, 
B 


1139. Slate Problems. 


1. A and B rented a field for a year for $175. A put in 6 
horses for the whole time, B put in 5 horses for 11 months and 3 
horses for 5 months. How much of the rent had each to pay ? 


2. A bankrupt surrenders property worth $1,287 for the 
benefit of three creditors to whom he owes $750, $1,125, and 
1,245, respectively. How much should each creditor receive? 


3. Four persons rented a pasture for 26 weeks. K put in 
50 sheep and L 60 sheep for the whole time, M put in 70 sheep 
for 20 weeks, and N 90 sheep for 22 weeks. How much of the 
rent, $180, had each to pay? 


REVIEW. 509 


4. A employs a capital of $2,500 in business, and at the end 
of 3 years takes into partnership B, who furnishes $4,000. Four 
years later they are joined by C, with a capital of $5,000. At 
the end of 12 years from the commencement of the business, the 
profits, amounting to $15,000, are divided. What amount 
should each receive ? 

A’s money is in the business how many years? B's, how many years? 
o's, how many? 

5. Four butchers rent a field, and pay for 6 months’ rent 
$152.50. The first puts in 20 oxen for 10 weeks and 50 sheep 
for 8 weeks; the second, 25 oxen for 8 weeks and 30 sheep for 7 
weeks; the third, 18 oxen for 10 weeks and 10 sheep for 12 
weeks; the fourth, 30 oxen for 12 weeks. What share will each 
have to pay, counting 3 sheep equal to 1 ox? 


6. A wall 700 yards long was to be built in 29 days. At 
the end of 11 days, 18 men had built 220 yards of it. How many 
extra men had then to be put to work, so that the wall might be 
completed in the given time? 


7. If 5 needlewomen can do a piece of work in 11 days of 9 
hours each, how long will it take 3 needlewomen to do two such 
pieces, supposing them to work 103 hours each day? 


8. If 14 men can mow 168 acres in 12 days of 8 hours 15 
minutes each, how many acres can 20 men mow in 11 days of 7 
hours 48 minutes each ? 


9. If 12 men can do a piece of work in 20 days, what num- 
ber of men will be required to do four times as much work in a 
fifth part of the time? 


10. A ship sailed with a crew of 60 men, and provisions for 
34 days, and 10 days afterwards, 12 persons were received on 
board from a sinking vessel. How long would the provisions 
last the 72 persons then on board? 


__ How long would the provisions last the 60 persons at the time the sink- 
ing vessel was met? 


510 


ARITHMETIC. 


11. If 76 boards, each 14 feet long and 10 inches wide, are 
worth $19.76, how much would 50 such boards be worth ? 


12. If 7 men receive $126 for 5 weeks’ work, how much 


should they receive for 9 weeks’ work? 


13. If for 7s. 6d. I can buy 9 Ib. of raisins, how many pounds 
ean I buy for £56 16s. ? 


14. A field of grain was to be cut down by 40 men in 10 days. 


Hight of the men, however, failed to come. 


take the others to do the work ? 


TABLE. 


Assessments. 


Real and 
Personal, 


For State 
Purposes. 


Tax Levies. 


For County 


Purposes. 


For City 
Purposes. 


How long did it 


1140. Brooklyn Assessments and Taxes for 10 years. 


of Valuation. 


rn fm nf | a | ES | 


¥ 298,936,506 |$ 874,088 


1883 
1884 
1885 
1886 
1887 
1888 


317,853,850 
330,683,762 
362,009,202 
383,851,674 
407,454,028 
428, 483,681 
452,758,601 
466,914,249 
483,738,129 


733,669 
889,559 
929,273 
907,663 
940,517 
1,344,023 
949,253 
589,178 
888,297 


1,323,861 
1,307,090 
1,412,623 
1,398,310 
1,682,120 
1,997,414 
2,009,518 
2,159,879 
2,240,613 


1,242,476 |$ 5,632,795 


6,287,462 
7,383,911 
7,180,990 
8,266,643 
8,503,581 
9,298,236 
8,709,541 
9 241,130 
10,324,617 


es ee eee ee eee ee 


Find for each year the total tax levy, and the tax rate in dol- 
lars, cents, and mills per $1,000 of assessed value. 

Find the average assessment per year; the average tax levy 
for state, county, and city purposes; and the average tax rate. 


SURFACES OF SOLIDS. 


SURFACES OF PRISMS AND CYLINDERS. 


1141. Slate Exercises, 


Nors.— The pupils should be encouraged to make cardboard 
models of the forms studied. 


1. Find the convex surface of a square prism, 
one side of its base being 4 inches and its height 
6 inches. Draw the development. 


Notr. — The convex surface is the surface exclusive of the 
bases. 


2. Find the convex surface of a triangular 
prism, each side of whose base measures 4 inches 
and whose altitude is 6 inches. Draw the devel- 
opment. 


3. Find the convex surface of an hexagonal prism, 
each side of its base being 4 inches and its altitude 6 
inches. Draw the development. 


4. Can you show that the convex surface of a 
prism is. found by multiplying the perimeter of the 
base by its altitude (height) ? 


5. Find the convex surface of a cylinder, the 
diameter of its base being 4 inches and its height 6 
inches. 


6. How do you find the entire surface of a prism 
or cylinder? 


7. What is the entire surface of a cube whose side 
is 7 inches? Of a cube whose side is x inches? 


8. The entire surface of a cube is 216 sq. in. What is the 


length of one side ? 


9. The convex surface of a cube is 144 sq. in. 
entire surface. 


‘sy lb ARITHMETIC, 


10. Find the entire surface of a square prism, one side of 
whose base measures 4 inches, and whose altitude is 6 inches. 

11. The convex surface of a square prism is 600 sq. ft., the 
altitude is 15 ft. What is the length of one side of the base? 

12. The entire surface of a square prism is 1,650 sq. in. One 
side of the base measures 15 inches. What is its convex surface ? 
What is its altitude? 

13. Find the entire surface of a square prism whose convex 
surface 1s 040 sq. in., and whose altitude is 15 inches. 

14. What is the entire surface of a cylinder whose base has a 
diameter of 1 foot, and whose altitude is 1 foot? 


SURFACES OF PYRAMIDS AND CONES. 


15. The convex surface of a square pyramid 
consists of how many equal triangles? Find the 
convex surface when one side of its base meas- 
ures 4 inches and its slant height (AX) 6 inches. 

Draw the development. 


16. The convex surface of a pyramid is equal 
to the perimeter of the base multiphed by what? 

17. Find the entire surface of the above pyramid. 

18. Calculate the entire surface of a square pyramid. whose 
slant height is 18 inches, the area of its base being 144 sq. in. 

19. Find the entire surface of a triangular pyramid whose 
three convex faces and the base are equilateral triangles, each 
side measuring 2 inches. 

20. Draw the developed convex surface of a 
cone, the diameter of whose base is 4 inches, and 
whose slant height is 6 inches. 

Calculate the convex surface. 


eX i 


21. How many square inches of paper would 
be required to cover the side and the base of a cone 6 inches in 
diameter at the base, and having a slant height of 10 inches? 


VOLUMES. 513 


22. Calculate the slant height of a cone whose 
altitude is 12 inches, the diameter of its base being 
10 inches. What is its convex surface? 


23. What is the entire surface of a cone, the 
diameter of whose base is 6 inches, and its slant 
height 10 inches? 

Draw the development. 


6 in. 24. A semi-circular piece 4 Ne 
of paper 6 inches in diam- 
eter is folded into a hollow 
cone (without overlapping). 
What will be the diameter 

AB of the mouth of the cone (the base)? What will be the 
slant height BC? 


C 


VOLUMES OF PRISMS AND PYRAMIDS. OF CYLINDERS 
AND OONES. 


1145. Slate Exercises, 


SuaGEstion. — Have the pupils construct of cardboard a hollow square 
prism of convenient size, and a pyramid having base and altitude respectively 
equal to those of the prism. Let them use sand or water to ascertain how 
many times the contents of the pyramid must be taken to exactly fill the 
prism. 

Volume of prism or cylinder = area of base x altitude. 
Volume of pyramid or cone = area of base x + altitude. 
1. Find the volume of a square pyramid, the area of the 
base being 9 square feet and the altitude 6 feet. 
2. What is the volume of a square pyramid whose altitude 
is 12 inches, one side of the base being 10 inches? 
3. The base of a prism is a triangle whose sides measure 3, 
4, and 5 inches respectively. Find the solidity, its altitude 
being 10 inches. : 
4. The base of a prism 19 feet high is a rectangle whose sides 
are 9 feet and 13 feet. How many cubic yards does it contain ? 


514 ARITHMETIC. 


5. Find the volume of a prism whose bases are equilateral 
triangles, each side being 4 ft., and the height of the prism being 
12 ft. 


6. How many cubic feet are there in 


a stone roller 6 ft. long, 8 ft. in circum- 
ference ? 


7. Find the volume of a cone whose 
altitude is 18 meters, diameter of base 6 
meters. 


8. How many gallons of oil (231 cu. in.) will fill a cylindrical 
tank 54 ft. high, radius of base 8 ft.? 


9. Measure accurately the interior dimensions of a quart or 
a pint cup, and calculate its volume. 


Note. — How many cubic inches in a quart, liquid measure? 


10. Measure the interior dimensions of a peck or a bushel, 
and calculate its volume. 


11. Pour a quart or a pint of water into a paper box having 
a rectangular base, and calculate the number of cubic inches of 
water in the box. 

What would be the depth of a quart of water in a box whose 
base measures 51 by 3 inches? 


LUMBER MEASURE. 


1147. Lumber is measured in board feet. A board foot is 1 
foot long, 1 foot wide, 1 inch thick. 

A board 16 feet long, 1 foot wide, 1 inch thick, contains 16 
board feet. 

A board 16 feet long, 9 inches wide, 1 inch thick, contains 
(16 x 3) board feet, or 12 board feet. A board of the same length 
and width, 2 inches thick, contains (12 x 2) board feet, or 24 
board feet. 

In practice, the term board foot is seldom used, the word foot 
alone being generally employed. 


LUMBER MEASURE. 


515 


1148. Find the number of feet (board feet) in each of the 


following boards and planks: 


16 feet long, 12 inches wide, 1 inch thick. 


ee = Se ee 
PF Ww NY HF OS 


15. 


CO Xe TP ww o 


14 
12 
14 
16 
12 
14 
16 
14 


Pae, 


12 


ce 


ce 


ce 


6 


iss 


ce 


ce 


1 


ep ~The) Seep (by ese [Ney Se) Sep ss) sy SS 


16. What is the cost, at $30 per thousand feet, of 15 planks, 


each 16 feet long, 9 inches wide, 3 inches thick ? 


17. Find the number of (board) feet of lumber required to 
floor a dock 36 feet long, 17 feet 6 inches wide, the planks being 
24 inches thick. 


18. Find the duty, at $1 per thousand feet, on the following 


lumber imported from Canada: 


13 feet long, 8 inches wide, 1 inch thick; 


150 boards, 
60 planks, 
40 scantlings, 15 feet long, 5 inches wide, 4 inches thick. 


14 feet long, 9 inches wide, 2 inches thick; 


516 , ARITHMETIC. 


19. At $18 per thousand, what will be the cost of the boards 
necessary to enclose a field 160 yards long, 120 yards wide, with 
an open fence 4 boards high, each board 6 inches wide, and 1 
inch thick ? 

MENSURATION. 


1150. Slate Problems. 


Area of circle = 4 circumference x 4 diameter. 


Area of sector = 4 arc x 4 diameter. 
1. Find the area of asemicircle whose radius = __-------. ‘ 
is 20 feet. ne i i 
2. How many square inches are contained in Le 
a sector of 60°, the radius of the circle being 15 \ ¥ A 
inches? ao 
6Q ° 


3. A square is inscribed in a circle 10 inches 
in diameter. Find its area. 

x = side of square, x?= area. Find 2? from the 
right-angled triangle, without finding the value 
pf te. 

4. What is the difference between the area of 
a circle of 10 inches diameter and that of the inscribed square ? 


5. The sides of the above inscribed square are chords of 
arcs of 90°. Find the length of an arc of 90°, and of its chord. 


6. A segment of a circle is that portion of the 


J 
surface included between an arc and itschord. Find © ti 
the area of a sector of 90° and the area of the seg- 
ment, the radius of the circle being 10 inches. 10 


7. Calculate the area of a circle whose radius is 1 inch. Of 
a circle whose radius is 2 inches. What is the ratio of the two 
areas ? 
8. What is the ratio between the area of a circle whose radius 
is 1 inch and that of a circle whose radius is 3 inches? 
The area of a circle = square of radius x ? 


MENSURATION. 517 


9. How many square yards are there in a circular walk, the 
radius, AJB, of the inner edge of walk being 10 
feet, and that of the outer edge, AC, being 15 


feet ? 54) 


(Find the difference between the area of a circle of 15 
ft. radius, and that of a circle of 10 ft. radius.) 


C 


10. A circular flower-bed 20 feet in diameter 
is surrounded by a walk 5 feet wide. How many square feet of 
surface does the walk contain ? 


(If you have to subtract 100 times 3.1416 from 225 times 3.1416, how can 
you shorten the work ?) 


11. How many square inches are there in the surface of a 
frame 3 inches wide, around a looking-glass 6 
inches in diameter? 


(Area = ? x 3.1416.) 
12: What is the ratio between the surface 


of the above frame and that of the looking- 
glass ? 


(Indicate operations and cancel.) 


13. What is the area of a walk 5 feet wide around the out- 
side of a square plot containing 400 sq. ft.? 

(What is the area of the large square, including 
the walk ?) . 

14. The outer edge of a walk 5 feet wide, 
surrounding a plot of ground, measures 120 
feet, the inner edge measures 80 feet. How g 


many square feet does the walk contain ? 
120 + 80 
2 


D 


(The “average” length of the walk is = 100 ft.; that is, its 


length measured on a line along the center of the walk.) 


15. Find the ratio between the area of a triangle whose sides 
measure 16, 30, and 384 feet, respectively, and the area of 
another whose sides are 32, 60, and 68 feet. 


51S ARITHMETIC. 


SURFACE OF SPHERE. 


1151. Take a wooden hemisphere and drive a tack into the 
center of its curved surface. Commencing at the tack, carefully 
wind a waxed cord about the curved surface, in the way a boy 
winds a top. When this surface is exactly covered, cut the cord. 


4 . Ps 


ea aK 


Wind the same cord around a tack driven into the plane sur- 
face of the base of the hemisphere, pressing it closely to the sur- 
face. When the latter is entirely covered, just one-half of the 
cord will be used. 

If a sphere is cut through in any direction, the section made 
will be a circle. The section formed when the sphere is cut 
through the center is called a great circle. 

The above experiment shows that the surface of the hemi- 
sphere is equal to that of two great circles of the same sphere. 


1152. The surface of a sphere is equal to that of four great 
circles. 

Since the surface of a great circle of the sphere is 4 diameter 
< 1 circumference, the surface of the sphere is 4 diameter x 4 
circumference < 4 = diameter of sphere X the circumference. 

Calling the radius of a circle #, and using the Greek letter 
aw instead of 3.1416, we have 

Diameter of circle = 2 R. 


Circumference of circle = 27f. 
Area of circle = 7R. (Lof2R x tof 27h.) 
Surface of sphere = 47’. 


CUBE ROOT. 519 


1153. Slate Exercises. 


16. ‘Find the surface of a sphere whose radius is 1 inch. 

Of a sphere whose diameter is 2 inches. 

Of a sphere whose circumference is 6.2832 inches. 

17. At 10 cents a square foot, what will be the cost of gilding 
a sphere 12 inches in diameter ? 

18. Find the ratio between the surface of a sphere 1 foot in 
diameter, and the convex surface of a cylinder 1 foot high, the 
diameter of the base 1 foot. 

19. What is the ratio between the surface of the above sphere 
and the entire surface of the cylinder? 

20. Find the surface of a sphere whose circumference is 20 
inches. 


CUBE ROOT. 


1155. To cube a number is to employ it three times as a 


factor. 
The cube of 4, written 4°, is 4 x 4 x 4, or 64. 
Find the cube of 1, 9, 6, 3, 5, 8, 2, 7. 


To find the cube root of a number is to find one of the three 
equal factors of the number. wee 
The cube root of 343, written 348, is 7. 


The cube of 25, 20 + 5, is equal to the following: 
We have seen (Art. 1031) that 
(20 + 5)? = 207+ 2 xk 20 x54 5 


Multiplying by 20+ 5 we have 
Product by 20 = 20? + 2x 20?x 5+ 20 x 5? 


Product by 5= 20? 5+ 2x 20x 5? 4 58 
(20 + 5)8 = 20°? + 3 x 20?x5+ 3x 20x 5? + 58 
which may be written in this way, 
20 + [(3 x 20) + (3 x 20 x 5) + 57] x 5. 


a 


520 


ARITHMETIC, 


1156. Extract the cube root of 15,625. 


We see by inspection that 
the cube root is between 20 
and 30; thatis, 20+. Sub- 
tract from 15,625 the cube of 
20, 8,000. The remainder, 
7,625, is equal to the second 
number multiplied by the sum 
of three times the square of 
the first (1,200), ete. Using 


(20)' = 


20 +5 


3 X 20? = 1,200 


&8X20x5 = 300 
5S= 25 
1,525 


15,625 
8,000 
7,625 


7,625 


remainder 


1,200 as a trial divisor, the second number is seen to be 6 or less. 

Taking 5 as the second number, we add to the 1,200 three times the 
product of the first and second (300), and the square of the second (25), 
making a total of 1,525. Multiplying this sum by the second number, we 
get 7,625, which is equal to the difference between 15,625 and 8,000. The 
second number is, therefore, 5, and the cube root of 15,625 is 25. 


110,592 
40+8 
110,592 
408 = 64,000 
3 x 40? = 4,800 46,592 
3x40 x8 = 960 
82 — 64 
5,824 46,592 
Ans. 48. 


658,503 


3 x 80? 


83 = 


3x80 xX7 = 


Tie 


19 200 
1,680 
49 


20,929 


Ans. 87. 


8 7 
658/503 
512 
146,503 


146,503 


In the last example we point off three places, beginning at the right, and 
find the greatest cube in the first period, placing its cube root as the first 


figure of the answer. 


1157. Find the cube root of the following: 


1. 2197 
2. 9,261 
3. 32,768 
4. 68,921 
5. 148,877 


6. 
ib 
8. 
9. § 

10. 


238,328 
421,875 
551,368 


512 
29 
1331 

27 


11. BYP, 
12. 8,375 
13; 1244 
14. 188% 
15. 5454 


MENSURATION. VAT 


VOLUME OF SPHERE. 


1158. Cut up a sphere (a round potato, for instance) into a number of ° 
small pieces, passing the knife in each case through the center of the sphere. 


Hach piece is a solid, having for its base a portion of the surface of the 
sphere, and for its altitude the radius of the sphere. 

When the pieces become very numerous, the base of each may be con- 
sidered a plane, and the solid a pyramid. The volume of each pyramid is 


equal to the base x} altitude; and the total volume of all, which is the 
volume of the sphere, is equal to the total surface of all the bases, which is 
the surface of the sphere, multiplied by 4 altitude, that is, + radius. 


Surface of sphere = 47 f’, 
therefore, volume of sphere =47f’? XL R=47rh’. 


1159. Slate Exercises. 
1. Find the volume of a sphere whose radius is 3 inches. 
2. Ifthe diameter of a sphere is 3 inches, what is its volume? 


3. What is the ratio between the volumes of two spheres 
whose diameters are 1 foot and 2 feet, respectively ? 


4. Find the ratio between the volume of a sphere 1 foot in 
diameter, and that of a cube whose side is 1 foot. 


522 ARITHMETIC. 


5. The radius of a sphere is 18 inches. What is the circum- 
ference of a great circle? The surface? The volume? 


6. What is the weight of an iron cannon-ball 12 inches in 
diameter, considering the weight of a cubic foot of water as 1,000 
ounces, and considering iron 7.5 times as heavy as water ? 


7. Find the ratio between the volume of a sphere 4 inches in 
diameter, and that of a cylinder 4 inches in altitude, radius of 
base 4 inches. 


Nors. — Indicate the volume of each, and cancel. 


s. A man has a cubical block of hard wood, its side measur- 
ing one foot, which he wishes made into a sphere one foot in 
diameter. What decimal part of the block is cut away ? 

The volume of the sphere is about what fraction of the volume 
of the cube? 


CUBE ROOT. 


1162. Find the cube root of 9,938,375. 


When the root contains 3 Mes i bo Sil 
more than two figures, con- 9'938!375 
tinue, as shown in the accom- 


panying example, taking for Si 
divisor three times the square 8X 20?= 1200 1988 

of the first two figures con. OX 20X1 = 60 

sidered as tens, plus three times 1+—= Lila 

the product of the first two 3 x 210? = 182300 6773875 
figures considered as tens by By BIO ae B50 

the third figure, plus the square Be OB 

of the third figure. i = 


135 470 677 375 


1163. Find the value of the following: 
1. 1,442,897 3. 3,723,875 5. V12.977875 
2. 1,906,624 4. »/39,651,821 6. 66.923416 


ANNUAL INTEREST. 523 


ANNUAL INTEREST. 
1171. Slate Problems. 
Derroit, Micu., June 1, 1890. 
Four years after date, without days of grace, I promise to pay 
to the order of Daniel W. Lawler, Six Hundred Dollars, value 
received, with annual interest at six per cent. 
$ 600%. GEORGE OXNARD. 


1. Find the amount due June 1, 1894, no payments of prin- 
cipal or interest having been made. 


1172. When the maker of a note fails to keep his contract to pay 
interest annually, the laws of some states, including Michigan, permit the 
collection of swmple interest on the deferred payments of interest. 


Principal, $600.00 
Interest, 4 years, at 6%, 144.00 
3 years’ interest, at 6%, on the Ist year’s interest, $36, 6.48 
PV te: ‘ Me oe) Boe te ap OCLs - 

ive a PME tet POR aan, ;: : 
Amount due June 1, 1894, $ 


2. Find the amount due, at 5%, for 5 years, on a note for 
$1,200, annual interest being unpaid. 


3. What is the amount of a note for $720, at 4 years, at 
4$%, annual interest unpaid after the first year? 


4. The maker of a note for $900, with annual interest at 
7%, makes the first and the second interest payments when due. 
How much will he owe at settlement, 6 years after the date of 
the note ? 

5. Find the difference between the amount due at 6% for 3 
years on a note for $300, annual interest unpaid, and the amount 
of the same sum placed at compound interest for the same time 
at the same rate. 

6. Find the amount due March 1, 1899, on a note for $500, 
dated March 1, 1893, with interest at 6%, annual interest 
unpaid after the third year. 


524 


ARITHMETIC. 


U.S. GOVERNMENT LANDS. 


1173. In surveying government lands, a line is run east and west, 
called the base line, and one perpendicular to it, called the principal 


meridian. 

Parallel lines are run north 
and south, and east and west, 6 
miles apart, forming squares, 
called townships. The row of 
townships adjoining the prin- 
cipal meridian is called Range 1 
East or West, according to its 
location. The row of townships 
north of the base line is called 
Township 1, North; the row 
above, Township 2, North, etc. 

The township in the diagram 
marked by a star (*) is designated 
3 T. 8, R. 2 HE. (third township 


south of base line, in the second range east of the 


Tadentaele pore] | 


“es | 


| 
{Teen [ote 
LSAM er y ADNVY 


TOWN HIP|4 | a NORTH ie 

7.0 ag Wis w fa) Be 9 aoa heats ioe ee] 
Price ae] we p> 
A Wee | ara a ion at eae 
QO; | O9}/9192!10 
m—-m—-m m——-m—-M——m 
o};n] — YNio;Fia 
BASE Pad LINE 


principal meridian). 


1174. A township, which contains 36 square miles, is divided into sec- 
tions one mile square, numbered as in the diagram, No. 1 being found at 


the northeast corner. 


TOWNSHIP 


2020 |2e|araofes 
ENCES EEE 


SIX MILES 
tS) 


Each section contains 640 acres. 


SECTION 
N 
ONE MILE 


= 
ONE MILE 


ONE MILE 
S 


1175. Sections are divided into half-sections (320 A.) and quarter-sec- 
tions (160 A.), and the latter are subdivided into half quarter-sections 
(80 A.) and quarter quarter-sections (40 A.), 


METRIC SYSTEM. 525 


1176. Slate Problems, 


1. Find the cost of the S.W. 4 of the N. 4 of sec. 13, T. 7 
N., R. 4 E., at $1.872 per acre. 


2. What will be the cost of fencing, at 75% per rod, the W. 
4 of the N.W. 4 of sec. 36? 


3. Mr. Thompson owns sec. 1, and his brother owns sec. 30 of 
the same township. What is the length of the shortest lne 
between the boundaries of the two farms? 


4. A road runs east and west between townships 4 and 5, 
south. Another road runs north and south between R. 7 and 8 
east. How far is it by road from the north-east corner of T.55., 
R. 10 W., to the north-west corner of T. 7 N., R. 8 E.? 


5. How many feet of boards, 6 inches wide, would be needed 
to build an open fence, 4 boards high, around the N. 4 of the 
S.W. 4 of sec. 16? 


6. The owner of secs. 19 and 20 has sold the W. 4 of N.W. 4 
of sec. 19; also the N. 4, the N. 4 of S8.H. 4, and the S.E. 4 of 
the 8.E. 1 of sec. 20. Draw a map of the land he still owns, and 
calculate its area. 


METRIC SYSTEM. 


1177. The metric system, which is used in nearly all the 
countries of continental Europe, is based upon the meter. The 
length of the meter is one ten-millionth part of the length of the 
meridian from the equator to the poles — about 39.37 inches. 


1178. The subdivisions of the meter are denoted by the Latin 
prefixes milli (z54,), centi (;1,), deci (7). For the multiples, 


the Greek prefixes deka (10), hecto (100), kilo (1,000), and 
myria (10,000) are used. 


1179. It will be nuwticed, in the table below, that small 
letters are used for the abbreviations of the Latin prefixes of the 


526 ARITHMETIC. 


subdivisions, and capital letters for the Greek prefixes of the 
multiples. 
The following is the table of 


1180. Measures of Length. 


10 millimeters (mm.) 1 centimeter (cm.) 
10 centimeters 1 decimeter (dm.) 
10 decimeters 1 meter (m.) 

10 meters 1 dekameter (Dm.) 
10 dekameters 1 hectometer (Hm.) 
10 hectometers 1 kilometer (Km.) 
10 kilometers 1 myriameter (Mm.) 


1181. The units of this table in common use are the centi- 
meter, the meter, and the kilometer. 


1182. A person who wishes to buy 124 meters of cloth, would not ask 
for 1 hectometer 2 dekameters 4 meters, any more than a New York mer- 
chant would tell a person who owes him $38.75 that his bill is 3 eagles 8 
dollars 7 dimes 5 cents. 


1183. Long distances are expressed in kilometers. The thickness of 
wire is given in millimeters. 


1184. Problems. 


1. What will be the cost in francs of 880 m. 75 of dress 
goods at 2 f. 60 per meter? (880.75 meters @ 2.60 francs.) 


2. How many square meters in a piece of carpet 26 m. 50 
long, 85 cm. wide? 


3. How many square meters in a circle whose diameter is 
15 meters? 


4. Anareisa surface 10 meters long, 10 meters wide. How 
many ares in a field 135 meters long, 69 meters wide? 


5. Find the area in ares of a right-angled triangle whose 
base is 245 meters, hypotenuse 875 meters. 


METRIC SYSTEM. Lapiyy 


6. A stere is a cubic meter. What will be the cost, at 
8 f. 50 per stere, of a pile of wood 10 meters long, 1 meter wide, 
3m. 25 high? 


7. A cube one decimeter each way contains a liter (1.), which 
is the principal unit of dry and liquid measure. 


How many liters’ capacity has a tank 10 m. 50 long, 8 m. wide, 
6 m. 50 high? 


8. How many bottles, each containing 0 1. 75, can be filled 
from a hogshead containing 222 1. ? 


9. How much will be received for 36 bags of beans, each con- 
taining 68 liters, at 1 mark 25 per dekaliter? 


10. A liter of water weighs a kilogram (1,000 grams). How 
many kilos of oil would a tank contain, its dimensions being 5 
meters X 4 meters x 3 meters, the weight of the oil being 92% 
of the weight of water? 


11. Assuming the length of the meter as 39.37 inches, what 
is the length of the kilometer ? 


1185. Greater accuracy is assured in operations requiring multiplication 
and division by indicating the operations beforehand, and performing the 
division last. 


Length of meter in yards =. 1 mile = 1,760 yd. 1 km, = 1,000 m. 


39.37 x 1,000 _ _ 3,937 
361,760 36x176 


12. Mt. Blanc is 4800 m. high. How many feet high is it? 


mile. 


Ans. = 


1186. In the following ten problems call the meter 40 inches. 
Give answer in two decimal places. 


13. How many cubic inches in a liter? (See problem 7.) 
How many quarts? 


14. How many bushels in a hectoliter? How many gallons? 


528 ARITHMETIC. 


15. How many pounds in a kilo, when a cubic foot of water 
weighs 1,000 oz.? (See problem 10.) 


16. What would be the circumference of the earth in miles if 
the meter measured 40 inches? 


(The meter is zggq5500 Of What part of circumference ?) 
17. How many square yards in a square meter? 
18. How many acres in a hectare? (See problem 4.) , 
19. How many rods in a hectometer ? 
20. How many cubic feet in a stere? (See problem 6.) 


21. How many troy grains (7,000 to ay. lb.) in a gram? 
(See problem 15.) 


22. How many kilometers in a mile? 


1187. Measures of Surface. 


100 sq. mm. = 1 sq. cm. 
100 sq. cm. =18q. dm. 
100 sq. dm. = 1 sq. m. = 1.196 sq. yd. 


1188. The square meter is the principal unit of surfaces, such as walls, 
ceilings, floors, etc. 
100 centiares (ca.) = 1 are (a.) = 119.6 sq. yd. 
100 ares = 1 hectare (Ha.) = 2.47 acres. 


1189. The are is the principal unit of surface of small plots of land. 
The area of a farm is expressed in hectares, of a country in square kilo- 
meters. 


1190. Measures of Volume. 


1,000 cuimm?==)1 cusem: 
1,000 cu. cm. =1 cu. dm. 
1,000 cu. dm. =1 cu. m. = 35.316 cu. ft. 


L191. The principal unit is the cubic meter. 


METRIC SYSTEM. 529 


1192. The stere (cubic meter) is used for measuring wood. 
10 decisteres (dst.) = 1 stere (st.) = 35.316 cu. ft. 
10 steres = 1 dekastere (Dst.) 


The stere is the only unit used. 


1193. Dry and Liquid Measures. 


10 milliliters = 1 centiliter 


10 centiliters =1 deciliter Dry. Liquid. 
10 deciliters =1 hter(l.) = .908 qt. = 1.057 qt. 
10 liters =I1dekaliter 1.1385 pk.= 2.642 gal. 


10 dekaliters =1hectoliter 2.837 bu. = 26.417 gal. 
10 hectoliters = 1 kiloliter 
10 kiloliters = 1 myrialiter 


1194. The liter and the hectoliter are the principal units. 


1195. Table of Weight. 


10 milligrams (mg.) 1 centigram 


10 centigrams 1 decigram 

10 decigrams 1 gram (gr.) 

10 grams 1 dekagram 

10 dekagrams 1 hectogram 

10 hectograms 1 kilogram (kilo) 2.2046 lb. 
10 kilograms (Kg.) 1 myriagram 

10 myriagrams 1 quintal 

10 quintals 1 tonneau (ton) 


1196. The kilo is the ordinary unit. Heavy articles are sold by the 
tonneau. 


CHAPTER XV. 


ALGEBRAIO EQUATIONS.—TWO UNKNOWN QUANTITIES, — 
THREE UNKNOWN QUANTITIES.—PURE QUADRATIOS, — 
AFFECTED QUADRATICS. | 


ADDITION OF ALGEBRAIC QUANTITIES. 


1199. Sight Exercises. 


Add: 
1. 2 fours 2. 6 hundredths 3. $4 4. 8¢ 5. Tx 
3 fours 8 hundredths $5 5¢ Ax 
4 fours 10 hundredths pT 8¢ 22 
5 fours 12 hundredths $8 9¢ 5a 
~? fours ~? hundredths $? o¢ 2a 
6. — 2a M+ 8e@. 8) —Say 9. Jabe) TO. BA aye 
— 4a + 42 —4 ry 15 abe — S2yz 
— 6a + 52 — wy 6 abe — xyz 
— Ta +102 — 22y abe — ld2yz 
—19a +? @ hey ? abe ht Ye 


1200. In the quantities 2a, 3x, 5xzy, l5abc, the numbers 2, 
3, 5, 15, are called coefficients. When no coefficient is expressed, 
1 is understood. Thus, abe =1labe. 

Where no sign is expressed, + is understood. 


1201. What a person has may be represented with a plus 
sign (+) placed before the amount; debts may be shown by a 
minus sign (—) placed before the amount. 

530 


ALGEBRAIC EQUATIONS. 531 


A has $500; B owes $300. If they unite their fortunes, what 
will they be worth together? 
4+. $500 
— $300 
+ $200 


Both together are worth $ 200. 
The sum of + 500 and — 300 is + 200. 


1202. If A had $3800 and B owed $500, the firm would be 


$200 in debt. 
(+ $3800) + (— $500) = — $200. 


1203. Add: 

1 —2a 2. Tx 8 —5ay 4. — 9Yabe &. — 24xyz 
—4a —4z —42y l5abe 5 xyz 
—6a — 2x xy 6 abe LY2 

7a 5 2 2 xy — abe 15 xyz 
—5a 62 —? xy if ? 


1204. Can you give the rule for addition where the quantities 
have different signs? Which sign does the sum take? 


1205. Add: 
6. 82+14, —7x#+9, — 23, 42—5, —2za, and 8x2+11. 
82+14 
—Tx+ 9 
— 23 
4x— 5 
—22 
_ 82+il 
7 4a+32, —2a, —Txa—3a, —52, —9a+ze. 
8. —8b+c, 4a+6), 55—9c, —8a, —2a—3b+4e. 
9. 42—8, —x2+4, —12—38, Tx+16, —5x—10. 
10. 42+ 23, —8x+ 241, —2x4+4+11, —x+5, 92-38. 


532 ARITHMETIC. 


SUBTRACTION OF ALGEBRAIC QUANTITIES. 


1206. Oral Problems. 


1. The thermometer in the morning was 33 degrees, at noon 
it was 52 degrees. What was the difference in temperature? 


2. In December the thermometer was 10 degrees below zero. 
In July it was 90 degrees above. What was the difference in 
temperature ? 


3. Two cities are in the same latitude. One is in 34° east 
longitude, and the other in 17° west longitude. What is their 
difference in longitude? 


4. What is the difference in longitude between two cities on 
the equator, one being in 56° west longitude, and the other in 
47° west longitude ? 


5. A boy makes 40¥ one day and 50f the next. How does 
he stand at the end of the two days? 


6. How would he stand if he made 40 one day and lost 50¢ 
the next day? 


7. A man traveled from the town M, 60 miles due north, 
and then traveled 50 miles due north. How far is he from his 
starting-point ? 


8. One day a man goes 50 miles due north; the next day he 
travels 70 miles due south. How far is he then from his starting- 
point? 


9. On Monday A is worth $250; on Tuesday he is worth 
$150. What has he lost in a day? 


10. A man has $150 Jan. 1. Feb. 1 he owes $250. What 


has he lost in a month? 


1207. The degrees above zero on a thermometer may be indi- 
cated by a plus sign (+); those below, by a minus sign (—). 


ALGEBRAIC EQUATIONS. 5338 


What is the difference between + 52° and + 33°? Between 
+ 90° and — 10°? 
Show by a diagram. 


1208. A has $600, B owes $400. What are they worth 
together ? 
(+$600) + (— $400) =? 
‘How much better off is A than B? 
(+ $600) — (— $400) =? 


1209. In subtracting algebraic quantities, change the signs of 
the subtrahend, and proceed as in addition. 


1. From 8a take 2a. 5. From — 8a take — 2a. 
8a 6. From — 2a take 8 a. 
—2a 
Ans. 6a 7. From — 2a take — 8a. 
2. From 2a take 8a. 8. From 2a take — 8a. 
2 
beam 9. From 82+ 14 take x+ 10. 
Ans. — 6a 82+14 
3. From — 8a take 2a. Seal 
— 8a 10. From 52—8 take —382—9, 
— 2a 
ee Oc 11. From 2 — 28 take 5a — 87. 
4. From 8a take — 2a. 12. From 7x+ 16 take 92 — 4. 
8a 13. From 62 take 22 —5. 
+ 2a 
Ans. 10a 14. From 82 take 9z+8. 


15. From 8zx+2a—5 take x—a—4Y. 
16. From 7y—22+6 take —8y+6b—z. 
17. From c—d-+e take ct+d—f. 


534 ARITHMETIC. 


REMOVING PARENTHESES. 


1210. From 8 take the difference between 49 and 25. 
84 — (49 — 25) = what? 
Would the result be the same if we should write the above 
84 — 49 — 25? 
What sign must be changed? 


1211. Write the following without parentheses: 

1. 57+ (83 — 16) = 74 4. (17—8)—(16— 14)=7 
2. 92— (63 + 25) = 4 5. 75+4x (15 —10)=95 
3. (48—10)+(24—5)=52 6. 75—4x (15—10)=55 


1212. Is there any change made in the signs of the first? In the signs 
of the second? Ofthe third? Ofthe fourth? Ofthe fifth? Of the sixth? 


1213. Solve the following equations. Prove the correctness 
of your answers. 


1. 6(22—5)=52+12 

NotE. 6(2%—5) means 6 times (2a —5), or 12a—30. 

2. T(¢+2)=382+50 4. 3(16 — 2x) = 4(18 — z) 

3. 53 +2)+16=61 5. 15(a —8) = 2(189 — 162) 
6. 38—(11—9xz)=10x 


Removing the parenthesis, we have 
| 38 —11+4+9a¢=10" 

Transposing, 9a—10%=—38+411 
or, —x2=—27 

Bringing — a to the right side of the equation, and — 27 to the left side, 
we have 

(+) 27=(+)# 

In practice, however, when the result is such as the above, — x = — 27, 

the signs of both members are changed, and the result becomes 


x= 27 


ALGEBRAIC EQUATIONS, 535 
7. 2(a —1)—2(2"—19)'= 3(# — 8) 
8. 6(22—5)—5x=12 
9. 5x —6(22—5) =~ 12 


Ou Onna 
EOC Tanner ae | 


Clear of fractions by multiplying both members of the equation 
by 10, and observe which sign must be changed to preserve the 
equality. 


1214. =2 


When x = 6, the above may be written 


B2—-6 42-4 
2 5 
Clearing of fractions, 


15 2— 30—(8x—8)=20 


=2 


Removing the parenthesis, 


15z—30—82#+ 8= 20 


Transposing, 15x—8xr=20+4 30—8 
or, (27=42 
z=6 


Notre. — The horizontal line between the numerator and the denominator 
of the foregoing fractions has the effect of a parenthesis, the entire quantity 
above the line being divided by the number below. 


18 24—4 
5 


o—* = (18 - 6) +2 1 of (24 — 4) 


at =(40—4) +5 


536 


1215. Solve: 


ii; 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 


25. 


ARITHMETIC. 


Tx—8 =z? _ z+2 
yee 


40—5ae 52492 


24=3-+4 2124—(5+422)+4 28 
§¢4+9=22+4+ (82-42) 


1 H 5 x x ae 
ARNG ce? 


f2—120=— +10 


r—20=(7+15)4 


Mae ad bk 
9(82 4+1)—-4=4(92+5)+3 


5a—6 
2 3 = -——— 
x-+ 5 


ALGEBRAIC EQUATIONS. 537 


1216. Slate Problems. 


1. A certain number is multiplied by 52; 7 is subtracted 
from the product; the remainder is divided by 16, giving a quo- 
tient of 3. What is the number? 


2. Three-eighths of what number is 60 less than the number 
itself? 


3. Four persons are of the same age. If the first were 4 of 
his age older, the second 4 of his age older, the third + of his age 
older, and the fourth 4 of his age older, the sum of their ages 
would be 99 years. What is the age of each? 


4. A man spends } of his earnings on board and lodging, + 
on clothing and repairs, and 4 on sundries. At the end of the 
year he has $280 left. What are his yearly earnings? 


oe, @ 
==+—+= + 280. 
(2 ras oa ) 


5. A boy gave 4 of his marbles to one companion, and 1} of 
them to another. He then bought 4 as many as he originally 
had, and had 4 marbles more than he had at first. How many 
did he have at first ? 


6. A father’s age and a son’s age added together amount to 
138 years. Twelve years ago the father was twice as old as the 
son. How old is each now? 


Let « =son’s age 12 years ago. 2a = father’s age then. 
7. John has 80 cents, and William has 60 cents. How many 


ane will William have to give John so that the latter shall have 
4. times as much money as the former? 


After William gives John a cents, the former has (60 — x) cents, and the 
latter has (80 + a) cents. 


8. In how many years will a man, now 25, be double the 
age of his 11-year-old brother? 


Let « = number of years. 25+ 2 and 1l+a= ages after x years. 


538 ARITHMETIC. 


9. A man has a cask of 60 gallons’ capacity. He draws off 
one-fourth of its contents, and then fills it. If it takes 24 gal- 
lons to fill it, how many gallons did the cask originally contain? 

10. A number is divided by 3, and 401s subtracted from the 
quotient, leaving a remainder of 104. What is the number? 

11. The difference between two numbers is 430. When the 
greater is divided by the less, the quotient is 4, and the remainder 
is 76. What are the numbers? 


Let x = less. greater 4, 76. 
less less 


12. A person pays $103 with 29 $2 and $5 bills. How many 
are there of each denomination? 


13. A father is 80 years older than his daughter. In 4 years, 
his age will be four times her age. What are their present ages? 

«and «+ 30=present ages. «+4 and a + 34=ages 4 years later. 

14. The product of two numbers is 180. If the smaller num- 


ber be increased by 8, the product of the two numbers will be 
225. What are the numbers? 


smaller = 2; WS = greater. 
15. A man’s wages are $1 per day more than his son's, 


For 88 days’ work, the father receives $12 more than. the son 
earns in 40 days. Find the wages of each. 


16. The sum of two numbers is 47; their difference is 17. 
What are the numbers? 


17. A mother is 41 years old. Her son’s age is 5. In how 
many years will the son’s age be 4 of his mother’s? 


TWO UNKNOWN QUANTITIES. 
1217. Preliminary Problems. 


1. I paid a dollar for two 25 balls and five bats. How much 
did I pay apiece for the latter? 


ALGEBRAIC EQUATIONS. 539 


2. When three times one number is added to five times 
another, the sum is 84. If the second number is 12, what is the 
first number ? 


3. A girl paid 75¢ for + pound of tea and 21 pounds of 
coffee. The coffee cost 20% per pound. What was the price of 
the tea per pound? 


4. A man sold pigs at $5 each and lambs at $8 each, receiv- 
ing $42. He sold 4 lambs. How many pigs did he sell? 


5. Four times a father’s age added to twice his daughter’s 
age amounts to 180 years. The girl is 10 years old. What is 
the father’s age? 


6. Eight peaches and seven pears cost 44%. The peaches 
cost 2¢ each. What is the cost of a pear? 


7. Two pieces of cloth and eleven pieces of silk contain 152 
yards. ‘There are 10 yards in each piece of cloth. How many 
yards in each piece of silk? 


8. Two-thirds of a yard of linen and three-fourths of a yard 
of lace cost 40%. The price of the lace is 82¢ a yard. Find the 
price of the linen. 


9. Three and one-half times one number added to four and 
one-third times a second number equals 60. The second number 
is 9. What is the first number? 


1218. Slate Exercises. 

Find the value of the unknown quantity : 

1. 8x+7y =44. When x= 2, find the value of y. 
2. 3y+ 52=34. Find the value of 2; y =8. 

Soar Lebo ten) 10 le eae 

4. l4¢+7y=98 w=31; y=? 

5. 24+ $2=40. 2=82. 


540 ARITHMETIC, 


9x2—25y=8. uses |? 
8hy +442 = 60. z= 9. 

162—192 = 49. z= 0. 

Ty— 82=18. y = 64. 
10. 8224+ 50y=2,600. y= 20. 


ei ee he 


1219. A boy gave 17% for 3 lemons and 4 oranges, another 
boy paid 25¢ for 3 lemons and 8 oranges. How much did the 
lemons cost apiece ? 


x = cost of lemons su+4y=17 (1) 
y = cost of oranges dx+8y= 25 (2) 
Subtracting (1) from (2) 4y=8 
The oranges cost 2¢ each y=2 


How much apiece was paid for the lemons? 


11. If 8 coats and 14 vests cost $78, and 2 coats and 14 vests, 
at the same rate, cost $66, how much does 1 coat cost? What is 
the price of a vest? 


12. Given 42+7y=53 (1) 


Qa+8y = 25 (2) 
to find the value of y. 


First multiply (2) by 2, making it 4z+6y=50. Why? 
13. What is the value of z in equation (1), when the value 


found for y is substituted therein? Substitute the same value 
for y in equation (2) and find the value of z. 


1220. Find the values of w and y in the following equations: 
14. #+ y=165. 2z2+3y = 38. 
15. 22+2y=30. 2+ 3y = 27. 
16. 2x4+3y=18. 4x+38y= 24. 
17. 22+3y=40. 38x2-+2y = 35. 


1221. Given ( 


ALGEBRAIC EQUATIONS. 541 


18. 72+5y= 82. 22+ 2y = 28. 
19. 52+9y=14 94+5y=14. 
20. 34+5y=17. 8xt2y=17. 


x+8y=46 


eye ae es a To find values of x and y. 


Multiply (1) by 7, T2+2ly = 3822 


(2) 7a— 4y= 22 Subtract. 
25 y = 300 
y= 12 


Substituting this value of y in (1), we have 


21. 


22. 


23. 


24. 


25. 


26. 


27. 


28. 


x +36 = 46 
x =46—36=10 
Answers.2 = 10) y= Laie 
x+y=18 Add or subtract. 
z—y= 4 
42+38y=17 (1) Multiply (2) by 2 and subtract. 
22— "y= 1 (2) 
82+4y=48 Add. 
x—4y= 0 
382+ 5y=18 (1) Multiply (1) by 7 and (2) by 3. 
Ta+3y=18 (2) Subtract. 
4x+5y= 32 Add. 
62—Sy=-—2 
382+4y=3 (1) Multiply (2) by 2. Add. 
122—2y=8 (2) 
52=6y+5 Transpose. 
32=d5y—4 . 
38xz+5y+ 8=0 29. y—2x=—8r—-1 
24%z— y—12=0 2y—4x2=y+2+9 


542 ARITHMETIC. 


30. ia aH 17 Clear of fractions. 


——-+ —4— 20 
4 x 8 
31. da+thy=42 35. 444+ 32y = 67 
datty=11% T4a—5hy=12 
32. 232—Ty= 8x£+51 36. 3( «+ 7)=9(y—9) 
lly=152+2 4(3 2 —8)=17y—155 
33. x+y = 100,000 37. 2(a—11)— 2(y—9)=6 
5a , 4y zr+9 32 
—— + —4+ — 4,640 =— 
100 ' 100 y—3 15 
3sa2+7 z—4,y-—1 
34, ——_=5 38. Z_—=5 
oY & 3 a 4 
(Becht ea ol | 
5y+3 3 4 
39. 2a+d5y+3_¢ 
3%—4y—2 
Bo a deel re ae 
x—2y+2 


1222. Slate Problems. 
1. The sum of two numbers is 87. Twice the first added to 
three times the second is 96. What are the numbers? 


(Let x = first number; y = second number.) 


2. The difference between two numbers is 28. Five times 

the first less twice the second is 197. What are the numbers? 
(a —y = 28; 5a—2y =197.) 

3. The product of the first of two numbers by 5, added to the 
product of the second by 3, gives 37. The product of the first 
by 6, diminished by 5 times the second, equals 10. Find the 
numbers. 

4. Divide 65 into two parts whose difference shall be 19. 

(Let x and y=parts. Solve also by one unknown quantity.) 


ALGEBRAIC EQUATIONS. 543 


5. A person pays $103 with 32 bills, some of them $2 bills, 
the others $5 bills. How many of each does he use? 


6. For 25 head of pigs and sheep, a farmer received $145. 
How many of each did he sell, if he sold the former at a each, 
the latter at $5 each? 


7. 10 oranges and 4 peaches cost 88%; 6 oranges and 7 
peaches cost 82¢. Find the cost of an orange. Of a peach. 


8. 5 pounds of tea and 3 pounds of coffee cost $3.75; 8 
pounds of tea and 1 pound of coffee cost $5.05. What is each 
worth per pound? 


9. A farmer buys a certain number of horses at $125 each, 
four times as many cows at $45 each, eight times as many sheep 
at $10 each, and half as many pigs at $5 each, spending $1,550 
for all. How many of each does he buy? 


10. A man paid 75¢ for 2 pounds of raisins and 8 pounds of 
cheese. 5 pounds of raisins and 2 pounds of cheese at the same 
price would have cost 94%. What did each cost per pound? 


11. The sum of two numbers is 19. The sum of the second 
number and ten times the first, minus the sum of the first and 
ten times the second, equals 45. What are the numbers? 


12. Reduce 5 to an equivalent fraction, the sum of whose 
numerator and denominator shall be 126. 
x=numerator; y = denominator. 
5 


4 
eae ~ 126. 
Oa T Wi 


13. What fraction equivalent to 38 has 147 for the difference 

between its numerator and denominator? 
(« —y=—147. Why?) 

14. 10 pounds of coffee at 830% per pound are mixed with z 
pounds of coffee at 25¢ per pound. What is x equal to, when 
the mixture is worth 26% per pound? 

25 x +(10 x 30) = 26 (10 +2). 


544 ARITHMETIC. 


15. A grocer mixes green tea costing 60% per pound with 
black tea costing 40% per pound. He uses 100 pounds in all, 
and the mixed tea costs him 48% per pound. How many pounds 
of each does he use? 


Let «=number of pounds of black tea; y= number of green. Then 
3+ y =number of pounds of mixed tea. 


z+y=100; 402+ 60y = 48 (x+y). 


THREE UNKNOWN QUANTITIES. 


1223. 1. Given the following: 
8z2+2y— z2=12 (a) 
o2—4y+3z2=16 (6) 
22+38y+22=385 (ec) 
to find the values of x, y, and z. 


(a) multiplied by 5, 15z+10y— 5z2=60 
(d) He AS 15*a—12y+ 92=48 
Subtract, 22y—14z2=12 (d) 
an equation containing only two unknown quantities. 
(6) multiplied by 2, 10zx— 8y+ 62= 382 
(c) i tO. 10z+15y+10z2¢= 175 
Subtract, —28y— 42=— 143 (e) 


an equation containing only two unknown quantities. 


Compare the two equations (d) and (e), which contain the 
same two unknown quantities. 


(d) multiplied by 2, 44y—282= 24 
(e) ‘ Kat — 16ly— 28z=—1,001 
Subtract, 205 y = 1,025 


y ea 


ALGEBRAIC EQUATIONS. 545 


Substituting this value of y in (d), we have 
110—14z=12, —14z=— 98, z=7. 
Substituting values of y and z in (a), we have 
38z+10—T=12, 324=9, x=8. 


Ans. += 8, 
y=5,| 
| teat 


2. Find the values of the unknown quantities in the following 
equations : 


z—8y+2z2= 3 (a) 
Qa+ yt+3z2= 22 (6) 
52+2y+72=851 (ec) 

Multiply (a) by 2, and subtract from (6). Multiply (a) by 5, 
and subtract from (¢). This gives two equations, each of which 
contains two unknown quantities. 

Compare these two resulting equations, and eliminate y. 

3. Se—2Zy+ z= 10 (a) 
382+ 8y—5z=120 (6) 
ta—3y—2z2= 8 (ce) 

Eliminate z by comparing (a) and (6), multiplying the forme 
by 5. Compare (a) and (c), multiplying the former by 2. 

4 182-— 4y4+15z= 317 
Tz+ 2y— 82= 89 
2la2—l1l7y+ 9z2z=—104 


& — 84+ y—12z2=— 259 
Tzaz— 4y+25z= 418 
182+ 2y—412=— 500 


ad 


546 ARITHMETIC. 


e. X42t¥_14 
AA 3 
ada Medea DUE GY 
2 6 : 
RO Ula Be eB ty 
2 5 
ly cA ah 
5 4 31 3 
By ue 10 anes 
8. 2+ 3 4y—32 
194.22 by 9 ore 
9. SE Dk fog diame 2 
4 4 6 
2e-bays Qari Sy 4 Oly aie yy 
2 8 4 16 


1224. Slate Problems. 

1. A man placed 3 of his capital at 5% and the other third 
at 6%. At the end of a year, capital and interest amounted to 
$31,600. What was his capital? 

2%, 5 x 


6 ‘ 
ox —— and 2x ——- = interest. 
3 * T00 an 3 * 00 intere 


2. A has 18 chestnuts more than B. If each finds 4 more, 
A will have four times as many as B. How many chestnuts has 
each ? 

3. Two mechanics earn together $8 per day. One works 23 
days and the other 17 days, for which they receive together $166. 
What does each earn per day? 


4. The sum of the first and the second of three numbers is 55, 
of the first and the third 62, of the second and the third 88. 
What are the numbers? 


s 
ALGEBRAIC EQUATIONS. 547 


5. The sum of two numbers is 53. Four times the first is 20 
more than twice the second. Find the numbers. 


6. A certain sum of money is divided among four persons. 
The first takes + of it, the second takes 4 of the remainder, the 
third takes 2 of what then remains, the fourth receives the 
balance, $24. What is the share of each of the other three? 


7. A merchant sold a lot of goods for $510, thereby losing 
3 of their cost. What did the goods cost ? 


8. A man collected a bill for a physician and deducted 5 
of the amount for his services, If he gave the physician $147, 
what was the amount collected ? 


9. Divide 1301 acres of land among three persons, giving 
the first 274 acres more than the second, and the second 133 
acres more than the third. 


10. A merchant has sold 4 of a piece of cloth, and has remain- 
ing 16 yards more than + of the piece. How many yards did 
the piece contain originally ? 


11. A servant is engaged for a year for $280 and a suit of 
clothes; he leaves at the end of six months, and receives $130 
and the suit. What is the value of the clothes? 


MULTIPLICATION OF ALGEBRAIC QUANTITIES. 


1225. Multiply 7+ 3 by x+4. 
The product is equal to x times (2 + 3) + 4 times (x + 3), 
“(r+ 3)=24+32 
4(4 + 3)= 47+12 
(c+ 3)(2+4)=27+72%+412 Ans. 


Nore. —-2” is read x square. The 2 is called an exponent. 


548 ARITHMETIC, 


Multiply (7+ 7) by («+ 8). 


z+iT 

x+8 
Product by 2, V+ 
Product by 8, 8x + 56 


2 +152+56 Ans. 
1226. Multiply: 
1. ( +5) by (4+ 2) 4. (22+8) by ( +9) 


2. ( +8) by (&+9) 5. (3%+1) by ( +7) 
8. (22+5) by (4+ 2) 6. (2%+1) by (22+1) 
1227. (x —5) x (w+4)=? 
x —5d 
zx +4 
x(a—5)  2—5e 
4(a— 5) 4x — 20 
x’? —xz—20 Ans. 


1228. Find products: 

Nore. — (w — 3) (w + 9) means « — 3 multiplied by x + 9. 
7. (e—38)(@+9) 10. (u+5)(e«—5) 13. (2xr—6)(8x2+8) 
8. («—6)(«x+7) 11. (2e—6)(a4+1) 14. (82+6)(22—3) 
9. (x—5)(a+5) 12. @—6)(2441) 15. (2x+43)(22—8) 


1229, (2 -—5)(x—4)=? 
The product is equal to 2(«—5)—4(x—5); that is, that 
4(2 — 5) is to be subtracted from a(x — 5). 
a(x —5)=2?—52; 4(a— 5) = 44 — 20. 
Placing the subtrahend under the minuend, and changing the 
signs of the former (Art. 1209), we have 
2 —52x 
— 42+ 20 
(x - 5)(a@— 4) = 2*?—927+20 Ans. 


ALGEBRAIC EQUATIONS. 549 


1230. (x —7)(x—9) =? 


Using either as a multiplier, place one under 
the other. Commencing with 2, say «x # = 2%, 
—9xa=—9a. Taking —7 as a multiplier, say 
2X(—7)=—72, (—9)X(—7)=63. Combin- 
ing, we get the product. 


x— 9 
x— 7 
v— 92 
ret halite aes ead ota 


Ans. 2? — 162+ 638 


1231. Note that the multiplication of a + (positive) quantity 
by a + (positive) quantity gives a + (positive) product; that 
(+) x (—) or (—) X (+) gives a — (negative) product; and that 


(—) X (—) gives a+ (positive) product. 
as follows: 


This is usually stated 


1232. Lrke signs produce +, and unlike signs produce —. 


1233. Give results: 
16. («—7)(x«—7) 20.( x+7)( x—6) 
17. («—5)(«@—9) 21.( x—4)( x—T7) 
18. (7+5)(@+5) 22. (2x—4)(8x—6) 
19. (x—3)(4+8) 23. (2%+6)(8x2—7) 


PURE QUADRATICS. 


2+6_ 32°— 66 


24. (2u+7)(82+8) 
25. (2x—3)(8 x— 2) 
26. (24—3)(20+38) 
27. (2x+9)(42—6) 


1234. Given ait to find the value of x. 
Clearing of fractions, 927+ 54=152* — 3380 
Transposing and combining, — 62? =— 884 
Dividing by 6, and changing signs, x’? = 64 
Extracting square root, z=+8. 


1235. Since (— 8) x (— 8) = 64, the square root of 64 may be 
either +8 or —8. It is written + 8, and is read “ posotwe or 
negative 8." (It is sometimes less correctly called plus or 


minus 8.) 


550 ARITHMETIC. 


1236. Slate Exercises. 

Find value of 2, y, z, etc. : 

1. 2? —13 = 36 11. (x — 3)(x+ 3) = 40 

2. 3y°+ 25 = 100 12. (7+ 5)(x+ 5)= 10x + 26 
3. 62-18 = 824 87 13. (24+ 4 =824 80 

4. 5(2’+17)—32°+63=198 44. 2+ 64—52 

5. 5(a?+17)—8(@'-21)=198 45. 3994 18-94 08 436 

6 
7 


seo ein ee 16. (28)? = (4 SP S18 
- (e+1)—2#'*=49 17. (x+-7)(a—9) =(x—8)(x—5) 
ae Pos 
la lh SOE since is. “44749 
3 a ee ee 
9 z+7 2—5 Ae et he 8 
z—38 2-9 “GE bi) 29 
to, 202 _. 80% PY fete Aaah ah 
z—l «+1 y-5 ytT 


1237. Slate Problems. 


1. Find the dimensions of a field, the length of which is 
éwice its breadth, its area being 1,800 square rods. 


2. The surface of the six equal faces of a cube contains 96 
square inches. Find the length of one edge. 


3. One number is fourth-fifths of another, and their product 
is 80. What are the numbers? 


4. One-third of a number multiplied by two-fifths of the 
same number gives a product of 270. Find the number. 


5. Thirty per cent of a number multiplied by forty per cent 
of the same number gives a product of 500. What is the num- 


ber ? 


ALGEBRAIC EQUATIONS. DoW 


6. Thirty per cent of forty per cent of a number is 300. 
What is the number ? 


7. The base of a right-angled triangle is $ as long as the 
perpendicular, and the area of the triangle is 96 square rods. 
Find the length of the base. What is the length of the hypote- 


nuse ? 

8. The base of a right-angled triangle measures x yd., the 
perpendicular measures sf yd. What is the length of the 
hypotenuse? If the hypotenuse measures 15 yd., find the length 
of the base. 


9. The base of a right-angled triangle measures x ft., the 
hypotenuse measures (7+ 9) ft., the perpendicular measures 15 
ft. What is the length of the base? 


10. The difference between the squares of two consecutive 
numbers is 49. What are the numbers? 


AFFECTED QUADRATICS. 


1238. Preliminary Exercises. 
(a@+1)\(@4+1]l=%7+2241 
The square of the sum of two quantities is equal to the square 


of the first + twice the product of the first and the second + the 
square of the second. 


(¢—1)(#-l=2—224+41 


The square of the difference of two quantities is equal to the 
square of the first — twice the product of the first and the second 
++ the square of the second. 


(a+b? =a¢+2ab+6 
(m—n? =m’—2mn+n' 
(10+ 5°? =10+2x10x5+8 
(10 — 3)? = 10?— 2x 10x 3+3? 


552 ARITHMETIC. 


1239. Oral Exercises. 


Square : 
1. 2+8 4. x<+10 
2. x—T7 5. a—b 


3. 2—9 6. z+y 


1240. Sight Exercises. 
Extract the square root of 
1. +6249 
2. 2 —1427+49 
3. 2?7—182+81 
4. 27+202z+100 
5. @+2ab4+0? 


1241. The square of (x + 3) consists of how many terms? Of 


how many terms does (# + 4)? consist? 


1242. Supply term necessary to make a complete square: 


1. 2?+62+? 
2. x2—122+? 
3. v2 —82r+? 
4. 2?—162+? 
5. 2?+1824+? 


1243. Slate Exercises. 


7 30—1 10. x-y 
8. 40—1 11. 80+5 
9. min 12. 60—5 

6. v@+2ay+y7 

7 @-—2Qryt+y¥ 

8. a—2ab+6 

9. 2? — 247+ 144 

10. 2’+222+4+121 

(2 + 5)?? 

Cir Aa ee 

7. 2 —4x+? 

8. 2?—1027+? 

9. #+1427+? 

10. 2?— 222+? 


Given vt+62=27 
What number must be added to the first member of the equa- 
tion to make it a ‘“‘complete’” square? 


If a number is added to one member of an equation, what 
must be done to the other member to preserve the equality ? 


ALGEBRAIC EQUATIONS. 5a 


1244. Extract the square root of .both members of the follow- 
ing equations, adding to both, where necessary, such a number 
as will make the first member a complete square. 


1 #+624+9=40+4+9 2. 2 —127%+ 36 = 28+ 36 


Remember that (+ 7) x (+ 7) = 49, and that (— 7) x (— 7)=49. 
- V49 =+ 7 or — 7, written +7. 


3. 2? —824+16=204 16 % 2@—l4a¢—15 
4. 2—162+ 64=— 39+ 64 pS Merkle ep ag hora! 


5. 2?+182+?=19+4+? 9. 2+142=51 
6. @+22+?= 244? 10. 2—222=48 
1245. Given xv*—102= 24. 


Completing the square, we have 2? — 10x+ 25= 24+ 25 = 49, 
Extracting the square root of both sides, we have 


z—5=+7, 
s=7+5=12 or —7+5=—2. 
Ans. 5 or — 2. 


1246. Find values of z: 


ele Oe | 9. 2—24x=0 
2. 2 122=108 10. at 82 = 384 
3. 2t1+22—48 Peo de 
4. 4182-115 12. 27+ 302—175 
5. 2@—l4x2=—— 13 13. 7+ 284 = 29 
6. 2?—10z2=0 14. 2?°+222=—104 
7. 2+202=— 125 15. 2? — 162 = — 64 
8. 27+ 262—56 16. a?+4+362= 76 


554 ARITHMETIC. 


1247. To make the first member a complete square, you 
added the square of what part of the coefficient of x? 


1248. Find values of z: 


Li ata el 5. 22+ 9x =— 20 
x -+ @ + ($) = 12+) 6. 2—llxz=— 28 
2. 2° —82=10 7. @+182=—42 
a?—32+($)=104+(3 ¢ 2 157=76 
3. 7 +52——4 9. 2?—-17z=18 
4. 2*°—iz=8 10. 27+19z=— 18 


1249. When 2’ has a coefficient, divide both members by the 
coefficient. 


3827+ 92 = 84. 
Dividing by 3, v’+3 a= 28. 
Completing the square, 
#4804 (r= 24+p=Sete Et 
Extracting square root, 2}+2—=—+41 
~a=ii—3—$=4; or —-i-—$=—-14=-—T7 
Ans. 4 or — 7. 
1. 62 — 62=36 . ot 9Ja=— 54 


6 
2. 943+ 92=180 % 827 — 722 —=— 160 
3. Ta?-+284—147 8. 727+ 492 —56 
4. 42% 402 —— 64 9. 32742le=54 
5. 827—162%—504 10. 5a?—25a=— 20 


ALGEBRAIC EQUATIONS. 555 


1250. Slate Problems. 


1. The sum of two numbers is 12; their product is 32. 
What are the numbers? 


wand 12—x2=numbers. (12 —.)«# = product. 


2. The base of a rectangle is 50 
feet longer than its altitude. Its area is 
2,400 square feet. How long is the 7 
base? 


Area 7? + 502 


2400 sq. ft. 


3. The perpendicular of a right- 
angled triangle measures 15 yards more than the base. 
The hypotenuse is 75 yards. Find the length of the 
perpendicular. 


1+ 2 


[ v2 + (15 + @)? = 75?.] 


4. The hypotenuse of a right-angled triangle is 1} 8” 
times as long as the base. The area of the triangle 
is 150 square yards. How long is the hypotenuse? 


[Perpendicular = V(3 a)? — a; area = 1 base x perpendicular. ] ay 


5. The entire surface of a square prism is 170 z 
square feet. Its altitude is 6 feet, and one 
side of its base is x feet. Find the value 50 +22 
of x. 
SP sd tt 
6. A garden 50 feet long, 40 feet wide, a S 


has a walk just outside it x feet wide. 
Find the area of the walk. 

If the area of the walk is 784 Sa 
feet, what is its width? 


7. A field, ABCD, contains 12 a 
acres. Its length is 1% times its e 
breadth. How many rods long is ; 
the diagonal BC? 


556 ARITHMETIC. 


8. A flag-staff, AB, 50 feet high, was broken 


13 
off at the point C. The broken part, resting on C, 
reached the ground JD, 30 feet from the base of ts 
the staff. Find the length of the part broken off. i 
9. A ladder, CH or DE, placed at a point &, is 
in a street 58 feet wide 
D_ between the opposite > 
C 
houses,. just touches &, ; 
. the top ofahouse, DB, 
3 60 feet high on one side es 33 
of the street, or the top of a house, 
A RB CA, 56 feet high on the other side. 
Find the length of the ladder. 
DE = 60+ (582) = Ch bb) 

10. A&C isatriangle. The side AB B 
measures 13 feet; the side BC, 4 feet; ue ¥ 
AC, 15 feet. Find the altitude BD. A iD eo 

Bie AR tT eer : 

11. ALCDisatrapezium. 4 B= 84 ft. ; 

BC= 20 ft.; CD=40 ft.; DA = 26 ft. ¥ E 5 


The perpendicular BF’ measures 16 ft. 
Find the length of the diagonal AC and 
of the perpendicular 4D. D 


CHAPTER XVI. 


ELEMENTARY GEOMETRY. — PROBLEMS IN CONSTRUC- 
TION. PRACTICAL APPLICATIONS. —CALCULATION OF 
HEIGHTS AND DISTANCES. — MENSURATION, 


ELEMENTARY GEOMETRY. 


1251. Angles, 

When two straight lines meet at a point, they are said to form 
an angle. 

The point at which the linesmeet _~— ia ae 
is called the vertex of the angle. 

When two angles are formed by the meeting of two straight 
lines, they are called adjacent angles. A 
and B are adjacent angles. C’ and D y / 
are adjacent angles. are / 

The angles #' and G, formed by the 
intersection of two straight lines, are called vertical, 
or opposite angles. F'and Hare vertical angles HO PH 
and #, F and G, G and A, H and #) are adjacent 


angles. 


When two adjacent angles 


are equal to each other, each ? oie 
is said to be a might angle. i OlP 
Thecanolesa lm yA. I, : 
NV, O, P are right angles. s 

An angle that is smaller than a right 
angle is called an acute angle; one larger /\ oe 
than a right angle is called an obtuse angle. a 
@ is an acute angle; £# is an obtuse angle. 

Angles that are not right angles are called, without regard 
to their size, oblique angles. 


557 


558 ARITHMETIC. 


1252. Designation of Angles. 

The angle formed by the lines SZ and 7'U may be called the 
angle 7. It is frequently better to call it 
the angle STU or UTS, the letter at the 
vertex being placed between the two others. Te U 

The use of the three letters is necessary 


where there is more than one angle having U 
its vertex at the same point, as in the accom- V 
panying figure, where UX, VX, and WX = 


meet at the point X. 


™R 


1253. Measurement of Angles. 


An angle is measured by the 
are of a circle, the center of the 
circle being at the vertex of the 
angle. The angle 123 is meas- 
ured by the arc YZ; the angle 
456 by the arc ab. 


1254. Circular Measure, 


60 seconds (/7) 1 minute, 
60 minutes (7) 1 degree, 
360 degrees (°) 1 circle. 


1255. The number of degrees in an arc or an angle is deter- 
mined by a protractor. 


A 
SEMI-CIRCULAR PROTRACTOR 


ELEMENTARY GEOMETRY. 559 


To measure an angle, XY YZ, for instance, produce the lines 
YX and YZ Place the point A of the pro- | x 
tractor on the-vertex ( Y) of the angle, and the 
edge AC’ on the line YZ produced. Using the 
lower line of figures, read off from the protractor 
the number of degrees at the point where the line YY produced 
cuts the semi-circle. 

In measuring the angle DEF’ the line AB " 
is placed on #//’ the point A on the vertex /. if 
The number of degrees in this case is read trom i 


the upper row of figures. 


LL 


EXERCISES IN CONSTRUCTION. 


1256. Norz.— In the following 100 exercises, the ruler, the compasses, 
and the protractor may be used. 
The drawing should be carefully done with a sharp, hard pencil. 


1. Draw an obtuse angle formed by two lines, each one inch 
long. Draw an acute angle formed by two lines, each six inches 
long. Which is the larger? 


2. Fold a piece of paper twice, so that the lines made by the 
creases will form four right angles. 

Fold a piece of paper so as to make four 4 
angles that are not right angles. 


3. The lines GH and JJ intersect at 
, making four right angles. Which arc 
is longer, 7 8, or ced? Which contains the 
greater number of degrees? 


4. Draw two lines meeting at an angle 
of 45°. Two lines meeting at an angle of 90°. Two meeting 
at an angle of 135°. 


5. Draw two lines making two angles, one of 
which measures 60°. How many degrees does 60° 
the other angle contain? 


560 ARITHMETIC. 


1257. Norr.—A line parallel to the right or the left side of the paper 
is called a vertical line; one parallel to Q 
the top or the bottom of the paper is called K Pp 
a horizontal line ; others are called oblique M N 
lines. 

KL is a vertical line, MN is a hori- R 
zontal line, OPand Qf are oblique lines. J, O 


6. Toa horizontal line draw a line making two equal adja- 
cent angles. How many degrees does each angle contain? 
To a vertical line draw a line making two equal adjacent 
angles. How many degrees does each angle contain ? 
To an oblique line draw a line making two equal adjacent 
angles. How many degrees does each angle contain ? 


7. How many degrees are there in a right angle? 


8. To an oblique line draw a line making two unequal adja- 
cent angles. How many degrees are there in the sum of the two 
angles ? 


9. How many degrees in the angle 7, if & ag 
s 
contains 75° ? 
(og 
V 


V measures 110°. How many degrees does U 
measure ? 

If one of two adjacent angles measures 634°, 
how many degrees are there in the other angle? 

How many degrees are there in an angle adjacent to one of 
47° 45'? 

10. Construct angle 5, 60°; angle 4, 50°. 
Measure angle 3. 

How many degrees and minutes will there be 3/4 
in angle 5, when 3 contains 494° and 4 contains 
833°? 

When angle 3 contains 36° 30’ and angle 5 contains 79° 45/, 
how many degrees and minutes will angle 4 contain? 

11. Erect a perpendicular at each extremity of a horizontal 
line. At each extremity of a vertical line. At each extremity 
of an oblique line. 


ELEMENTARY GEOMETRY. 561 


1258. Norr.—A line making a right angle with another line is said 
to be perpendicular to it. 
12. Construct a square upon a horizontal line. Upon an 


oblique line. ie 


13. Draw two lines intersecting at an angle 
of 100°. Mark in each of the other three angles 
the number of degrees it contains. 


14. If one of the four angles formed by two intersecting lines 
measures 90°, what does each of the other three measure? 
If one measures 60°, what does each of the others measure ? 


_ 15. At each extremity of a horizontal line draw a line making 
an angle of 40° with the first line. 


16. At each extremity of a vertical line draw a line making 
an angle of 100° with the first line. 


17. At one extremity of an oblique line draw a line making 
with the first line an acute angle; at the other extremity draw 
a line making an obtuse angle with the first line. 


18. Draw two lines making an angle (6) of 


150°. Construct an adjoining angle (7) contain- <i 
ing 80°. How many degrees will angle 8 con- 
tain? 


19. How many degrees will there be in the 
sum of five angles having the same vertex? 


20. Draw five equal angles having a common 
vertex. 
Is any one of these five angles adjacent to any other? 


21. Draw six equal angles having a common vertex. Is any 
angle adjacent to the angle next it? Why? 
Are any of the angles vertical? Why? 


22. Draw two angles, one of 65° and the other of 25°. Draw 
a third angle equal to the sum of both. 
Draw an angle equal to their difference. 


562 ARITHMETIC. 


23. Draw an angle equal to the sum of three angles measur- 
ing, respectively, 40°, 50°, and 60°. 


24. How many degrees are contained in the angle made by 
the hour and the minute hand of a watch at 1 o'clock? How 
many degrees in the angle made by the hands of a church-tower 
clock at the same hour? 


25. How many degrees are passed over by the minute hand 
in a quarter of an hour? How many are passed over by the 
hour hand in an hour? In half an hour? In 15 minutes? 

What angle is made by the hands at 12:15? At 6:30? At 
8:20? 


1259. Norr.— The pupils should, of course, understand that the angles 
in questions 24 and 25 are those formed by imaginary lines passing through 
the centers of the hands. They should know, too, that a geometrical line 
has neither breadth nor thickness 


1260. Parallels. 


Lines lying in the same plane that will not meet, no matter 
how far they are produced, are said to be parallel. 


26. Draw two or more lines that shall be perpendicular to a 
horizontal ine. Where will they meet? 

Draw two or more that shall be perpendicular to a vertical 
line. Where will they meet? 

Draw two or more that shall be perpendicular to an oblique 
line. Where will they meet? | 


27. To a horizontal line draw two or more lines running in 
the same direction, and each making an angle of 35° with the 
first line. Will the oblique lines meet? 

Draw two or more lines running in the same direction, and 
each making an angle of 125° with a vertical line. Will the 
oblique lines meet if produced very far? 

Draw two or more lines running in the same direction, and 
each making an angle of 74° with an oblique line? Will the 
former lines meet? 


ELEMENTARY GEOMETRY. 563 


28. Draw two lines making angles of 30° and 60°, respectively, 
with a third line. Will the two former lines meet if produced 
in either direction? 


29. Draw a line, AS, meeting a hori- A 
zontal line BC at an angle of 58°. Draw 
a third line, D#, parallel to the horizontal 
line, and cutting the oblique line. What 
angles does it make with the oblique line? 

Draw a fourth line, #’G, parallel to the 
oblique line, and cutting both horizontal lines. 

Mark in each of the twelve angles the 
number of degrees it contains. 

Take a stiff card, having a square corner, and 
cut off the corner, making the side AB equal to 
the side BC, and each about 4 or 5 inches long. 

Measure the angle at A and the angle at C. 

30. With a ruler, draw a line D#; and without moving the 
ruler, draw lines /'G, HT, RF Ht yt} 

AB, by placing the side Ds 
BC of the triangle firmly 
against the edge of the 

ruler. 

FG, HI, and AB are D—& 
parallel. Why? 


31. Holding the triangle against the edge of ruler JK, draw 
oblique lines AC, LM, WO. A v6 N 

If the angle ACB contains | 
45°, how many degrees will 
IMJ contain? How many 
will WOT contain? 

Are AC, L&M, and NO 
parallel? Why? 

32. Draw a line making an angle of 70° with 
a horizontal line. At the point P, using only 
the ruler and the triangle, draw a line parallel / . 

: : 70 

to the oblique line. P 


564 ARITHMETIC. 


33. Wh and UV are parallel lines, 
cut by a line SZ’ If the angle 4 meas- 
ures 50°, how many degrees does a 
measure ? 

Find the number of degrees in each of 
the other six angles. 


34. Draw WX and YZ intersecting 
at any angle. With the ruler and the , 
triangle draw a line parallel to WX. 

Mark all the angles that are equal 
to 7. Mark all those that are equal 
to 7. 


1261. Triangles. 


35. From the extremities of the line AB, draw lines that shall 
make angles of 60° and 40°, respectively, with 


AB. Prolong the lines until they meet at C, q 
forming a triangle. 
Measure the angle at C. How many , o\p 


degrees does it contain? How many degrees 
are there in the sum of the three angles of the triangle? 


36. Construct a triangle having one angle of 90° ete one of 
30°. Measure the third angle. 

How many degrees are there in the sum 0 
of the three angles? AG 


37. Draw a triangle having angles at the 
base measuring, respectively, 57° and 68°. . 
With the ruler and the triangle draw JJ ~-—— —--—J 
parallel to D&. Produce DH to A and 
FE to G. 

How many degrees will the angle a con- ee 
tain? How many degrees in the angle c? a 
Calculate the number of degrees in the angle 6 when you know 


ELEMENTARY GEOMETRY. 565 


the number of degrees in a and ¢, respectively. What angle is ver- 
tical to 6? How many degrees in d? 


38. Construct a triangle, ALM, 
making the angles at the base 28° 
and 120°, respectively. Draw, as be- 
fore, VO, parallel to LI. 

Is the angle e equal to any angle of 
the triangle? How many degrees does it contain? Is the angle 
Ff equal to any angle of the triangle? How many degrees does 
it contain ? 

How many degrees are there in the sum of the angles e, g, and 
f? How many degrees are there in the angle g? 


39. The angle at P is 70°, the angle at Q P 
is 60°. Can you tell how many degrees there ) 
are in the angle PAS, formed by producing 
the base RS? Q Rain 


The angle PRS is called an exterior angle of the triangle PQR. 


40. How many degrees are there in the three angles of any 
triangle ? 

41. Two angles of a triangle measure 36° and 65°, respectively. 
How many degrees does the third angle contain? 


1262. A triangle containing a right angle is called a right- 
angled triangle. 

A triangle containing an obtuse angle is called an obtuse- 
angled triangle. 

A triangle all of whose angles are acute, is called an acute-angled 
triangle. 

Obtuse-angled triangles and acute-angled triangles are also 
called oblique-angled triangles. 


42. Draw a triangle that shall contain an acute angle. Mark 
the acute angle. 

Draw one that shall contain two acute angles. One that shall 
contain three acute angles. 


566 ARITHMETIC. 


43. See if you can draw a triangle containing three right 
angles. 

A triangle containing two right angles. 

A triangle containing one right angle. 


44. Try to draw a triangle that shall contain three obtuse 
angles. 

A triangle that shall contain two obtuse angles. 

A triangle containing one obtuse angle. 


45. Draw a triangle containing two angles of 50° and 70°, 
respectively. How many degrees are there in the third angle? 

Measure each side, and mark on the side its length. 

Opposite which angle is found the longest side? Opposite 
which, the shortest side? 


46. Draw a triangle having two angles of 75° each. Are any 
two of its sides equal ? 

Draw a triangle having two angles of 50° each. Are any of 
its sides equal ? 


47. Draw a triangle having two angles of 60° each. How 
many degrees does the third angle contain? 
Are any of its sides equal? 


1263. A triangle having all its sides equal, is called an egu- 
lateral triangle. 

A triangle having two equal sides, is called an 2sosceles triangle. 

A triangle having all its sides unequal, is called a scalene 
triangle. 


48. Fold evenly on the line AC (Fig. 1), a rectangular sheet of 
paper. Cut off, on the line AJB, one of the 
folded corners. When this piece is opened out 4 
(Fig. 2), it makes what kind of a triangle? 
Which are the equal sides? How does the 
crease line, AC, divide the base, BA? 90° os 

Ifthe angle ABC, Fig. 1,is 65°, andthe angle 9% 
at Cis aright angle, what is the angle BAC? 


ELEMENTARY GEOMETRY. 567 


How many degrees in the angle BAB', Fig. 2? AL 
How does a perpendicular let fall upon the base 
of an isosceles triangle from the opposite angle 
divide the angle? How does it divide the base? 
How do the angles at the base of an isosceles BY’ C 2B 
triangle compare with each other as to size? peat 


1264. The wnequal side of an isosceles triangle is called the base. 


49. Draw an isosceles triangle having the base a vertical line. 
An isosceles triangle having the vertex below the base. 
One having an oblique line for the base. 


50. Draw a right-angled isosceles triangle. How many 
degrees will there be in each of the other angles? 

Draw an obtuse-angled isosceles triangle. 

51. How many degrees will there be in each angle of an 
equilateral triangle ? 

Draw an equilateral triangle having one side vertical. 

Draw an equilateral triangle having its p fe E 
vertex below the base. 

52. DEF is an isosceles triangle, DF’ 
and EF’ being the equal sides. If the 
angle 1 measures 50°, how many degrees 
are there in each of the other five angles, 
when the line F/G bisects the base ? 


an to 
e & 


53. If the angle DF'# in the above zr 
triangle measures 45°, how many degrees will there be in the 
angles 1, 2, 3, 4, 5, and 6? A 


54. ABC is a right-angled tri- 
angle, the angle at 6 measuring 
90°, and the angle at C’ measuring 
30°. If the line AX is so drawn 
as to make the angle AXB equal [go 4 
to 60°, find the number of degreesin B xX 
the angles m,n, and p, respectively ? 


5 


568 ARITHMETIC. 


1265. Quadrilaterals. 


A plane figure of four sides 1s called a quadrilateral. 

When the opposite sides are parallel, the quadrilateral is 
ealled a parallelogram. (Fig. 1 to 8.) 

A rectangle is a parallelogram all of whose angles are right 
angles. (Fig. 1 to 4.) 

When the four sides of a rectangle are equal to each other, it 
is called a square. (Fig. 1 and 2.) 

The term oblong is frequently applied to rectangles whose 
adjacent sides are unequal. (Fig. 3 and 4.) 


Fia. 1. FiG: 2; Fia. 3. Fre, 4. 


A rhombus is a parallelogram all of whose sides are equal, but 
whose angles are oblique. (Fig. 5 and 6.) 

When the adjacent sides of a parallelogram are unequal and 
the angles are oblique, it is called a rhomboid. (Fig. 7 and 8.) 


a /  / Sees 


Fra. 5, Fia. 6. Pigret: Fia. 8. 


A trapezoid is a quadrilateral having only two of its sides 
parallel. (Fig. 9 and 10.) 

A trapezium is a quadrilateral having no two sides parallel. 
(Fig. 11 and 12.) 


ELEMENTARY GEOMETRY. 569 


1266. The altetude of a parallelogram is the perpendicular 
distance between its base and the side opposite. 


T B B xX HH ye5 Bank) 


The altitude of a triangle is the perpendicular distance between 
the vertex and the base, or between the vertex and the base 
produced. 

AB 1s the altitude of WANT; DX is the altitude of DBE; 
Gi o.G aT 


55. Draw a parallelogram. How many angles does it con- 
tain? Into how few triangles can you divide a parallelogram ? 
How many degrees are there in the sum of the angles of each 
triangle? How many degrees are there in the sum of the angles 
of a parallelogram ? 


56. Construct a parallelogram, the adjacent sides of which 
shall measure 2 inches and 38 inches, respectively, and the angle 
between them 60°. How long will each of the other two sides 
be? Measure each of the other angles. How many degrees are 
there in the sum of the four angles? 


57. Construct a trapezoid having a base of 5 inches, altitude 
3 inches, the angles at the base measuring 90° and 60°, respec- 
tively. Measure the remaining angles, and find the sum of the 
four angles. How long is each of the remaining sides? 


58. Fold a piece of paper twice at right angles, and cut off 
the folded corner, making a rhombus when the part cut off is 
opened out. 

Can you cut out a rhombus having two angles of 60° each? 
A rhombus having two angles of 80° each? 


59. Can you so cut a piece of paper, folded twice at right 
angles, that the part cut off will be a square? 


570 ARITHMETIC, 


60. We have seen that an equilateral triangle is egwiangular; 
that is, that all its angles are equal. Are the four angles of a 
square equal? Are the four angles of a rhombus equal? 


61. A triangle that has three equal angles is eguelateral; that 
is, all its sides are equal. Can you draw an equiangular paral- 
lelogram that will be equilateral? Can you draw an equiangular 
parallelogram that will not be equilateral ? 


62. Draw a rectangle, base 24 inches, altitude 2 inches. 
A rhomboid, base 24 inches, altitude 2 inches. 


63. Make, out of paper, a rectangle and a rhomboid, each 
having the above dimensions, and endeavor to ascertain, by cut- 
ting, whether or not they are equal to each other in area. 


64. Show, by cutting, that any parallelogram can be divided 
into two equal triangles. 


65. Make three rhomboids of different shapes, the base of 
each to measure 3 inches, and the altitude 2 inches. Are they 
equal to each other in area? 


66. Make three rhomboids of different shapes, the base of 
each to measure 3 inches, and the adjacent side 2 inches. Are 
they equal to each other in area? 


67. How do we calculate the area of a rectangle 3 inches by 
2 inches? Of a rhombus whose base is 8 inches, altitude 2 
inches? Of a rhomboid, base 3 inches, altitude 2 inches? Of a 
right-angled triangle, base 3 inches, altitude 2 inches? Of an 
obtuse-angled triangle, base 3 inches, altitude 2 inches? Of 
an acute-angled triangle, base 3 inches, altitude 2 inches? Draw 
each of the foregoing. 


68. Draw a rectangle having two adjacent sides measuring 3 
inches and 2 inches, respectively. A rhomboid having two adja- 
cent sides measuring 8 inches and 2 inches, respectively. A 
trapezium having two adjacent sides measuring 38 inches and 2 
inches, respectively. A trapezoid having two adjacent sides 


ELEMENTARY GEOMETRY. 571 


measuring 3 inches and 2 inches, respectively. A right-angled 
triangle, an obtuse-angled triangle, and an acute-angled triangle, 
each having two adjacent sides measuring 3 inches and 2 inches, 
respectively. 

Calculate the area of each, making such measurements as may 
be necessary. 


69. Draw three trapezoids of different shapes, whose parallel 
sides shall measure 3 inches and 24 inches, respectively, and 
whose altitude shall be 2 inches. 

Draw a rectangle 2 inches high that shall be equal in area to 
each of the foregoing. How long will be its base? What will 
be the relation of the length of its base to the lengths of the 
parallel sides of the trapezoids? 


70. Draw three trapeziums of different shapes, each having 
one diagonal of 3 inches, the perpendiculars let fall on this 
diagonal from the opposite corners measuring 24 and 2 inches, 
respectively. 

What is the altitude of a rectangle that is equal in area to 
each of the foregoing, when the base of the rectangle measures 3 
inches? 


1267. The Circle. 


71. With the compass points one inch apart, draw a circle. 
Draw a radius. How long is it? 


The line made by the compass pencil is called the circumference. A line 
drawn from the center to the circumference is called a radius. 


72. Draw a circle, radius 14 inches. From a point on the cir- 
cumference, draw a line through the center to an opposite point 
on the circumference. 

This line is called a diameter. How long is the diameter? 


73. Draw a circle, radius 12 inches. Draw a diameter and 
several radii. 
Write diameter, cureumference, radius, each in its proper place. 


572 ARITHMETIC. 


74. Draw a circle. Between two points on the circumference 
draw a line that does not pass through the center. 
This line is called a chord. 


75. Draw a circle. In it draw two diameters, a radius, and 
three chords. Write on each line its name. 


76. Draw a part of the circumference of a circle greater than 
one-half of it. Draw the chord. 


A part of the circumference is called an are. 


77. Draw an arc less than a semi-circumference. Draw a 
chord. Write the name on each. 
Can you make a chord that will be longer than the diameter? 


78. Draw two equal circles. In the first draw the chord of 
an arc of 120°. In the second, the chord of an are of 240°. 
What is the ratio between the two chords you have drawn? 


79. Ina circle draw a chord equal in length to the radius. 
How: many degrees are there in the arc whose chord has been 
drawn? 

80. Draw an arc of 72°. To its extremities draw two radii. 

The part of the surface of a circle enclosed by two radii and the inter- 
cepted arc is called a sector. 

81. Draw a sector of 60° (a sextant). A sector of Oi 
quadrant). 

82. Draw an arc of 120 degrees. Draw the chord. 

The part of the surface of a circle Senate by an are and its 
chord is called a segment. 

83. Draw several circles having the same center, but a 
unequal radii (concentric circles). 


84. Draw two equal circles just touching each other (tangent). 
Draw two unequal circles tangent to each other. 

Within a large circle draw a smaller one tangent to it. 

85. Draw circles of equal radii cutting each other. Draw 
intersecting circles of unequal radii. 


ELEMENTARY GEOMETRY. His 


1268. Pentagons, Hexagons, Octagons. 


86. Divide the circumference of a circle into four equal ares. 
Draw the chords, forming an inscribed square. 


87. If you wish to inscribe in a circle a figure of 
five equal sides, into how many equal arcs must 
the circumference be divided? How many degrees 
will each are contain? 


1269. A plane figure bounded by straight lines is called a polygon. 

A five-sided polygon is called a pentagon; one of six sides, a hexagon; of 
seven, a heptagon; of eight, an octagon; of nine, a nonagon; of ten, a 
decagon ; etc. 

A regular polygon is one that is both equilateral and equiangular. 


88. Inscribe a regular pentagon in a circle. Use the pro- 
tractor, 


89. Inscribeinacircle a regular hexagon. <A regular octagon. 
An equilateral triangle. 


90. Inscribe in acircle a regular hexagon. Connect the oppo- 
site corners by lines passing through the center of 
the circle, forming six triangles. 
How many degrees are there in each of the six | 
angles about the center of the circle? In each of \t 
the twelve angles at the circumference? How 
many degrees are there in the sum of angles 1 and 2? 
Is each of the six triangles scalene, equilateral, or isosceles ? 


91. Divide a regular inscribed pentagon into five equal tri- 
angles by lines drawn from the center of the circle. 

What kind of triangles are formed; isosceles, scalene, or equi- 
lateral ? 

How many degrees are there in each angle at the center? In 
each angle at the circumference? How many degrees are there 
in the sum of two adjoining angles at the circumference? 


574 ARITHMETIC. 


92. Circumscribe a square about a circle. An equilateral 
triangle. A regular pentagon. A regular hexagon. A regular 
octagon. 

93. Draw a pentagon. What is the smallest number of tri- 
angles into which a pentagon can be divided? How many 
degrees are there in all of these triangles? How many degrees 
are there in each angle of a regular pentagon ? 


94. Draw a hexagon. What is the smallest number of tri- 
angles into which it can be divided? How many degrees do 
the six angles of a hexagon contain? How many degrees are 
there in each angle of a regular hexagon? 


95. A quadrilateral is divisible into how few triangles? A 
pentagon into how few? A hexagon? A heptagon? An octa- 
gon? ) 

The smallest number of triangles into which a polygon is 
divisible is how many less than the number of its sides? 


96. How many degrees are there in each angle of a regular 
octagon ? 

97. Using a protractor, construct a regular pentagon on a line 
two inches long. 

98. On a three-inch line construct an equilateral triangle. 
On the same line construct a square, a regular pentagon, a 
regular hexagon, heptagon, octagon, nonagon, etc. 


99. Inscribe a square in a circle. Circumscribe a square 
about the same circle by drawing lines touching the four corners 
of the inscribed square. 

What is the ratio between the areas of the two squares? 


100. Inscribe an equilateral triangle in a circle. Circumscribe 
an equilateral triangle about the same circle by drawing lines 
touching the vertex of each of the three angles of the inscribed 
triangle. 

What is the ratio between a side of the inscribed triangle and 
a side of the circumscribed triangle? What is the ratio between 
the areas of the two triangles? 


ELEMENTARY GEOMETRY. SVs) 


PROBLEMS IN CONSTRUCTION. 


1270. In drawing the following one hundred exercises, only the ruler 
and the compasses are to be used. Use neither the protractor nor the tri- 
angle. 

1. Draw a circle, radius an inch and a half. Outside of it, 
and tangent to it, draw a second circle of an inch radius. 

How far apart are the centers? 


2. Draw two tangent circles having radii of an inch and a 
half and an inch, respectively, one within the other. 
How long is the line joining the centers? 


3. With centers 3 inches apart draw two equal circles tan- 
gent to each other. How long is the radius of each ? 


4. With centers three inches apart draw two equal circles of 
2 inches radius. Connect the centers. 

Draw a line joining the two points in which the circles inter- 
sect. How does this line divide the line connecting the centers ? 
Draw radii from each center to each point of intersection. 

5. Construct an isosceles triangle, base 3 inches, equal sides 
2 inches. 

Norr.— Use circles or arcs where necessary. 

6. Construct an isosceles triangle, base 34 inches, equal sides 
4 inches. 

Divide it into two equal parts. Do not locate the center of 
the base by measurements. 

7. Ona vertical line construct an isosceles triangle. With- 
out measuring the length of the base draw a perpendicular to 
the center of the base. 

8. Bisect a vertical line. An oblique line. 

Do not measure the length of the line. 


9. Construct an equilateral triangle on a two-inch line. 


10. Construct an equilateral triangle on a vertical line. On 
an oblique line. 


576 ARITHMETIC. 


11. Construct a scalene triangle. 
A triangle having sides measuring 1, 14, 2 inches, respectively. 
One whose sides measure 2, 24, and 3 inches, respectively. 


12. Can you construct an isosceles triangle whose base meas- 
ures 4 inches, equal sides 2 inches? 

Try to construct a scalene triangle with sides measuring 1, 2, 
and 3 inches, respectively. 


13. Draw acircle. In it draw a chord. 

Bisect the chord, using as few lines and as short ones as you 
can. 

Notre. — Do not use the ruler to ascertain the length of the chord before 
bisecting it. 

14. Divide a sector into two equal parts. 


15. Draw a circle. Draw achord. Draw a radius through 
the center of the chord. 
Is the radius perpendicular to the chord? Why? 


16. Bisect the are of a circle and its chord. | 
Bisect the arc of a circle without drawing the chord. 


17. Draw a perpendicular to the middle point of a horizontal 
line. To the middle point of a vertical line. To the middle 
point of an oblique line. . 


18. Draw in a circle two diameters perpendicular to each 
other. 


19. Divide the circumference of a circle into four equal parts. 
Into eight equal parts. 
Inscribe a square in a circle. 


20. Inscribe a regular octagon in a circle. 


21. Connect the opposite vertices of a regular octagon in- 
scribed in a circle by lines passing through the center of the 
circle. 


Lines connecting the opposite vertices of a polygon are called diagonals. 


ELEMENTARY GEOMETRY. 577 


22. Inscribe a square in a circle. Circumscribe a: square 
whose sides shall be perpendicular to the diagonals of the inscribed 
square. 


23. Construct an equilateral triangle on a horizontal line 1 
inch long. On the right side as a base, construct a second equi- 
lateral triangle. On the left side of the first triangle, construct 
a third. Construct three more, completing the hexagon. 


24. Can you circumscribe a circle about the above hexagon ? 
What is the radius of the circle? 


25. Inscribe a regular hexagon in a circle whose radius is 14 
inch. What is the length of each side of the hexagon ? 


26. Inscribe in a circle an equilateral triangle. .On each of 
its three sides construct an equilateral triangle. 


27. Construct an arc of 60°. Draw two lines meeting at an 
angle of 60°. 


28. Bisect an arc of 60°. Draw two lines meeting at an 
angle of 30°. 


29. Construct an angle of 60° and an angle of 30°. Draw two 
lines making an angle equal to the sum of the two angles first 
constructed. 


30. Erect a perpendicular at the end of a horizontal line. 
At the end of a vertical line. At the end of an oblique line. 


31. Construct an angle of 45°. An angle of 221°. An angle 
of 135°. An angle of 15°. An angle of 75°. 


32. Draw a circle, radius 1 inch. Draw a diameter, and pro- 
duce it an inch beyond the circumference. At the center of the 
circle erect a perpendicular to the diameter. 


33. An inch from one end of a 38-inch line, erect a perpendic- 
ular, using as few and as short lines as,possible. 


34. Hrect a perpendicular at the center of an oblique line. 
Erect a perpendicular at one end of an oblique line. Erect a 


578 ARITHMETIC. 


perpendicular at a point in a line between one end and the 
center. 


35. Construct a right-angled triangle, base 2 inches, altitude 
2 inches. 
A right-angled triangle, base 3 inches, altitude 24 inches. 


36. Ona line 2 inches long, construct a square. Construct a 
rectangle, base 3 inches, altitude 2 inches. 


37. On an oblique line 3 inches long, construct a square. On 
an oblique line 3 inches long, construct a rectangle, altitude 2} 
inches. 


38. Construct a rhombus whose side is 3 inches, altitude 24 
inches. 
A rhombus whose side is 8 inches, and which contains an 


angle of 60°. 


39. Construct a rhomboid whose adjacent sides measure 4 
inches and 3 inches, respectively, altitude 24 inches. 


40. Construct a right-angled isosceles triangle. An isosceles 
triangle containing an angle of 120°. One containing an angle 
of 185°. 


41. Construct an isosceles triangle, base 3 inches, altitude 3 
inches. Can you construct an equilateral triangle whose altitude 
shall be 3 inches? 


42. Construct a scalene triangle, base 3 inches, altitude 3 
inches. An obtuse-angled triangle having a 3-inch base, and the 
altitude 3 inches. 


43. Construct a triangle whose base measures 5 inches, the 
other sides being 3 inches and 4 inches, respectively. Draw its 
altitude geometrically. 


44. Construct a triangle whose base measures 4 inches, the 
angles at the base measuring 120° and 30°, respectively. Draw . 
its altitude geometrically, 


ELEMENTARY GEOMETRY. 579 


45. Drawa line. From a point above the line, let fall a per- 
pendicular to the line. 


46. Inscribe a regular hexagon in a circle. Divide it into six 
equilateral triangles by diagonals. 

How many degrees are there in each of the six angles at the 
center of the circle? How many degrees are there in the arc on 
which each angle at the center stands? 


47. Inscribe an equilateral triangle in a circle. Mark in each 
angle the number of degrees it contains. Mark on each arc the 
number of degrees it contains. 

An angle at the circumference of a circle is measured by what 
part of the arc on which it stands? 


48. Do you know how many degrees there are in each angle 
of a regular hexagon ? 

Construct a regular hexagon without drawing a circle and 
without using triangles. 


49. Construct a regular octagon, each side one inch. Do not 
draw a circle. 


50. If an angle at the circumference is measured by one- 
half of the are on which it stands, how many degrees are con- 
tained between two lines forming a right angle whose vertex is 
at the circumference ? 


51. Inscribe several right angles in a circle, their sides inter- 
cepting the same arc. 


52. Inscribe in a circle a right-angled isosceles triangle. 


53. Construct a square whose diagonal measures 4 inches. 
A rhombus whose side is 3 inches and which has one diagonal 
3 inches long. 


54. Inscribe in a circle a scalene triangle. How many de- 
grees are there in the sum of the arcs intercepted by the sides 
forming the three angles? 

How many degrees are there in the sum of the three angles? 


580 ARITHMETIC. 


55. Draw two equilateral triangles having one side common 
to both triangles. What figure have you drawn? 


56. Construct two triangles, each having two sides measur- 
ing 2 inches and 24 inches, respectively, and the angle made by 
these sides (the included angle) measuring 60°. 

Cut out the triangles. Place one upon the other, and ascertain 
if the third side of one triangle is equal to the third side of the 
other. Are the two remaining angles of the first triangle equal 
to the corresponding angles of the second ? 


57. Construct a triangle having a base of 3 inches, and angles 
at the base containing 60° and 45°, respectively. Construct a 
second triangle having angles of 60° and 45°, respectively, and 
the included side 3 inches. 

Cut both out of paper, and test the equality of the correspond- 
ing parts of each. 


58. Construct a triangle whose angles measure 30°, 60°, and 
90°, respectively. Can you construct another triangle having 
angles equal to those in the first triangle, but with sides not 
equal to those of the first triangle? 


59. Construct a triangle having sides of 2, 24, and 3 inches, 
respectively. 

Try to construct a second triangle having sides equal to those 
of the foregoing triangle, but having the corresponding angles 
unequal. 


60. Can you construct a triangle containing three angles of 
75° each? 

61. In a circle, inscribe a right-angled triangle whose hypot- 
enuse is 3 inches. 

One whose hypotenuse is 8 inches, base 2 inches. 

Hypotenuse 3 inches, perpendicular 14 inches. 

What is the diameter of the circle in each case? 


62. Draw two equal circles. Mark off an arc in one. Mark 
off an equal arc in the other. 


ELEMENTARY GEOMETRY. 581 


63. Draw two lines meeting at any angle. Construct a second 
angle equal to the first. 


64. Draw a horizontal line. From each end, and from two 
points between, draw four oblique lines that shall be parallel to 
each other. 

From one end of a three-inch horizontal line, draw an oblique 
line running upwards; at the other end, draw a line running 
downwards that shail be parallel to the first oblique line. 


65. Ona three-inch line as a diagonal, construct a rhomboid 
having two opposite sides each 24 inches long. Mark off on 
each of these two sides half-inch divisions. Draw lines through 
the diagonal, dividing the rhomboid into 5 equal parts. How 
do these lines divide the diagonal ? 


66. Divide a 38-inch line into 5 equal parts, 
Draw a line exactly 3 inch in length. One exactly # inch. 


67. Draw a right-angled triangle whose base is 4 inches, per- 
pendicular 3 inches. How long is the hypotenuse? 

One whose base is 3 inches, perpendicular 2 inches. How long 
is the hypotenuse? 

One whose hypotenuse is 4 inches, base 3 inches. How long 
is the perpendicular ? 


68. Can you draw a line measuring exactly V13 inches? 
One measuring exactly 7 inches? One measuring exactly V5 
inches ? 


69. Divide a 2-inch equilateral triangle into two right-angled 
triangles. Mark the number of degrees in each of the six angles. 

In each right-angled triangle, find the ratio between the length 
of a side opposite an angle of 30° and the length of a side oppo- 
site an angle of 90°. 


70. Construct a right-angled triangle, hypotenuse 4 inches, 
angle at base 80°. Measure the length of the perpendicular. 
Construct a right-angled triangle whose perpendicular is 14 


582 ARITHMETIC. 


inches, the angle formed by the perpendicular and the hypot- 
enuse being 60°. How long is the hypotenuse? 


71. Draw in a circle whose radius is two inches, the chord of 
an arc of 60°. Draw the chord of an arc of 180°. What is the 
ratio between their respective lengths? 

What is the ratio between the chord of an arc of 60° and the 
chord of an are of 800°? 


72. In a circle of 2 inches radius, draw a chord. At one 
extremity erect a perpendicular. From the other extremity 
draw a diameter. Where will this diameter meet the perpen- 
dicular? Why? 


73. Make a triangle of splints, fastening them by one tack at 
each angle. Will the triangle retain its shape under pressure ? 

Make a rectangle of splints. Fasten each corner by a single 
tack. Will the rectangle retain its shape under pressure? 


74. Ina circle draw three parallel chords the same distance 
apart. 
Draw three parallel chords not equidistant. 


75. At the circumference of a circle, draw a line perpendicular 
to a radius. This perpendicular is tangent to the circumference. 

A line that touches the circumference in a single point is called a tangent. 
A line that cuts the circumference is called a secant. 


76. Inscribe a circle in a square. 


77. Ina circle draw two chords not parallel. Bisect each by 
a perpendicular. Where will these perpendiculars intersect ? 


78. Inscribe a circle in an equilateral triangle. In an isosceles 
triangle. In a scalene triangle. 


79. Circumscribe a circle about an equilateral triangle. About 
an isosceles triangle. About a scalene triangle. 


80. Draw a circle, using a coin or something similar. Can 
you find the center? 


ELEMENTARY GEOMETRY. 583 


81. Draw an obtuse-angled triangle. Cireumscribe a circle 
about it. 


82. Without using the compasses, draw an are. Can you find 
the center of the circle of which the arc forms a part? 


83. Construct a rhombus whose sides are 4 inches, the acute 
angle being 30°. What is its altitude ? 


84. Construct a square on a line 3 inches long, using only 
ares of 3 inches radius. 


85. Can you construct a triangle with sides 2, 4, and 6 inches? 


86. Draw a circle, radius 8 inches. Draw an equilateral 
triangle around it. 


87. Construct an equilateral triangle, altitude 14 inches. Con- 
struct one having an altitude twice as great. 


88. Construct a triangle whose sides measure 1, 14, and 2 
inches, respectively. Construct a second having its correspond- 
ing sides twice as large as those of the first triangle. 

Construct a triangle whose angles measure 30°, 60°, and 90°, 
respectively. Try to construct a second triangle having its 


corresponding angles twice as large as those of the first triangle. 


89. Construct a square which shall be equal in area to two 
squares, one having a side of 2 inches, the other having a side of 
3 inches. 


90. Can you construct a square whose area will be 13 square 
inches ? 


91. Construct a square whose area will be equal to the dif- 
ference of area between two squares, one liaving a side of 8 inches, 
the other having a side of 2 inches. 


92. Construct a square whose side is 3 inches. Construct 
another having double the area. 


93. Construct an equilateral triangle, side 2 inches. How 
many 1-inch equilateral triangles can be made from it? 


584. ARITHMETIC. 


94. How many l-inch equilateral triangles can be made from 
an equilateral triangle whose sides are 3 inches? 
From one whose sides are 4 inches? 5 inches? 


95. Construct a triangle, sides 2, 8, 4 inches. Divide it into 
four equal triangles. Give the dimensions of each of the latter. 


96. Construct a triangle 1,14, 2 inches. Make a triangle 
nine times as large as the first by producing two of the sides 
and drawing a fourth line. What are the dimensions of the 
second triangle ? 


97. Draw a circle of 1 inch radius. Tangent to it, and en- 
closing it, draw one having four times the area of the first. 


98. Can you draw a circle having half the area of a circle 
whose radius is 2 inches? 


99. Draw an equilateral triangle, side 2 inches. Construct 
another of half the area of the first. 


100. Can you tell the ratio between the area of an inscribed 
and that of a circumscribed hexagon? 


EQUAL TRIANGLES. EQUIVALENT TRIANGLES. 


1271. Norr.—The protractor and the triangle may be used in the 
following exercises. 


1. Draw arectangle, base 24 inches, altitude 2 inches. Draw 
a rhomboid, base 21 inches, altitude 2 inches. Find the area of 
each. 


2. With a base 2} inches, altitude 2 inches, draw 
(a) A right-angled triangle. 

(6) An isosceles triangle. 

(c) One or more acute-angled scalene triangles. 
(d) One or more obtuse-angled triangles. 


Calculate the area of each. 


ss — 


ELEMENTARY GEOMETRY. 585 


3. Can you show, by cutting from paper, that a right-angled 
triangle having its base and perpendicular 4 inches and 3 inches, 
respectively, has the same surface as an acute-angled triangle 
whose base and altitude are 4 inches and 3 inches, respectively, 
and an obtuse-angled triangle whose base and altitude are 4 
inches and 3 inches, respectively ? 

Two triangles that have the same area are called equivalent triangles ; 
those having their corresponding sides and angles equal, each to each, are 
called equal triangles. 


4. Construct a triangle whose sides measure 14, 2, and 24 
inches, respectively. Construct another triangle having its sides 
of the same lengths. Are the angles of the second equal to the 
angles of the first? Are the triangles equal ? 


5. Draw two triangles each of which has two sides measuring 
14 and 3 inches, respectively, and the included angle 60 degrees. 
Is the third side of one triangle equal to the third side of the 
other? Are the remaining angles of the first triangle equal to 
the remaining angles of the second ? 


6. With a base 2 inches long, and with angles at the base 
measuring 50° and 60°, respectively, construct a triangle. Con- 
struct a second triangle having a base measuring 2 inches, and 
angles at the base measuring 60° and 50°, respectively. Are the 
two triangles equal ? 


7. A person wishing to ascertain the length, AJB, of a pond, 
places a pole at a convenient point, C, visible 
from A and ££. The distance BC is meas- 
ured, and a pole is set up, on a line with B 
and C, at D, the distance CD being made 
equal to LC. A pole is also placed at #, on 
a line with A and C, the distance CE’ being 
made equal to AC. 

Can you show that the length, AB, of the pond can be ascer- 
tained by measuring the distance DA’? 


586 ARITHMETIC. 


SIMILAR TRIANGLES. 


1272. Construct a right-angled tri- _ 0 
angle ABC. Make AS 4 inches, AC 
5 inches, BC 8 inches. 

At the points n, y, and s, distant from 
A one, two, and three inches, respec- m, 
tively, erect perpendiculars. Measure 7 A 
the length of each. ThE ay ang 


8. An is one-fourth of AB; ascertain the ratio between 


mn and CB. 

Compare xy and CB; see if oe ea 

slp ek 

As is three-fourths of AB; is rs =3BC? 

Measure Am, Az, and Ar. How does each compare with 
AC? 

Would the relations as found above, exist if. mn were not per- 
pendicular to AB? 

How do the angles in the triangle Amn compare with the 
angles in the triangle ABC? 


9. Draw a triangle whose sides measure 4, 5, and 6 inches, 
respectively. Cut out of paper a triangle whose sides measure 
2, 24, and 38 inches, respectively. Place an angle of the small 
triangle on the corresponding angle of the large triangle, and 
compare their respective sizes. 

How does the area of the small triangle compare with the area 
of the large triangle? 


10. Two angles of a triangle measure about 37° and 53°, 
respectively. The sides opposite those angles measure 3 inches 
and 4 inches, respectively. How many degrees does the third 
angle contain? Calculate the length of the third side. 

What will be the dimensions of a similar triangle whose area 
is one-fourth that of the given triangle? Give the approximate 
number of degrees in each angle of the small triangle. 


ELEMENTARY GEOMETRY. 587 


11. Draw a line DH, 27 inches long. At any angle draw 
DF. Commencing at D, mark off on DF’ quarter-inch portions, 
Teg. Ao. OLG.,1b0 9). Join OF, 
and with a triangle and a 
ruler draw 8h, 7g, 6/f, etc., 
parallel to9 #. Measure HA. 

Is it equal to hg, of, fe, etc.? P 
Why? 

Into how many parts is the line HD divided? What frac- 

tion of an inch does each part contain ? 


In locating the points 1, 2, 3, 4, etc., is it necessary to make the divisions 
+ inch? Would it be sufficient to use the compasses with the points any 
convenient distance apart? 


12. Divide a line 27 inches long into 5 equal parts. 


CALCULATING HEIGHTS AND DISTANCES, 


1273. To verify the results obtained by calculation, the pupil should 
make diagrams, drawing the figures to a convenient scale. 
1. If AB in a right-angled C 
triangle measures 120 feet, and a8 
a perpendicular, vw, erected 10 . 
feet from A measures 5 feet, 


calculate the length of BC. w- 
Aw:AB::wv: BC, 

: A W B 

1.€. TOs Ae Be 


2. A post 6 feet above ground throws a shadow of 74 feet. 


588 ARITHMETIC. 


3. Wishing to ascertain the 
distance between two houses, AR 
and S, on opposite sides of a 
stream, I measure a line SV, 
at right angles to SA, 200 feet. 
At JZ, 90 feet from S, the per- 
pendicular ZW measures 60 
feet. Required the distance SR. 


Vie LW VSS VT = VS — ST. 


4. Beginning at B, 100 feet from the bank of a river, a line, 
BC, is measured 1,200 feet long. At D, distant from C’ 50 
feet, the perpendicular DF is found 
to measure 90 feet. What is the s== 
distance from # to A, a tree on 
the opposite bank? How wide is 
the river? 


5. A boy, whoseeye(#)is4 feet . 7B Dae 
from the ground, can just see the 
top (A) of a steeple when he stands back 3 feet from a fence 
(CG) 6 feet high. The distance from the foot of the fence to 
the center of the base of the steeple is 177 feet. Find the 
height of the steeple AB. 


CD Vie arg LD CLs aie A Pe 


A 


ELEMENTARY GEOMETRY. 589 


6. Wishing to ascertain the distance AB, I measure a line, 
AD), at right angles to A, 12 chains; 
DE, at right angles to AD, 5 chains; 
and find that a line sighted from £ to 
B intersects AD at C, distant from D 
3.25 chains. What is the distance from 
A to B? 


Nore. — The triangles DCE and ACB are 
similar. Why? 


7. Wishing to find the height of a 
tower ft, 1 set up a pole, cd, 12 feet long above the ground. 
Another pole ad, 44 feet above ground, 1s set up at such a distance. 
that the tops of the two f 
poles and of the tower are 
in a line. The distance 
between the poles (ae or 
db) is 104 feet. The dis- 
tance from d to the foot 
of the tower is 195 feet. 
The width of the tower 
(kj) 1s 80 feet. 

The similar triangles aec and ahf give us the proportion ae: ah:: ec: hf. 

What is the distance ec? ah=bi=bdidk+h. kh=ikhj. When fh 
is found, what must be added to get the height of the tower? 


8. To determine the height of a building, IN, a person 
attached a strip of wood, ad (a tin 
tube or a piece of narrow pipe 
would be better), to a post, OP, 
in such a manner that sighting 
from a, he could just see J, the 
top of the building. He then O 
sighted down from 6, and marked a 
on the ground the point &, on a R 
line with ad. a * 

PQ was found by measurement to be 4 feet, RP 6 feet, PV 
120 feet. Required ALN. 


M 


590 ARITHMETIC. 


9. Wood-choppers, desiring to 
know the height of a tree before 
cutting it, sometimes make an isos- 
celes right-angled triangle of wood 
or paper, and “step off” the distance 
on level ground from the point 
at which they find they can just 
see the top of the tree looking along 
the hypotenuse of the triangle, the G~~ 
base being parallel to the ground. 


How high is the tree AL, if AC is 36 paces of 8 oF 
feet each, and the angle ACB is 45°? Mee 


10. £B is a point on the bank of a stream 
due east of A on the other bank. A boy 
walks due south of A until he reaches a point 
at which he finds, from his pocket compass, 
that he is directly south-west of 6. If the 
distance AC’ measures 119 yards, how wide is 
the stream ? 


1274. Miscellaneous Exercises. 


1. Calculate the length in inches of an arc of 60°, the radius’ 
of the circle being two inches. Calculate the length of an arc of 
120°. Of 180°. Of 240°. Of 800°. 


2. Calculate the length in inches of a chord of 60° in the 
above circle. Of a chord of 120°. Of a chord of 180°. Of a 
chord of 240°. Of a chord of 800°. 


3. In the parallelogram shown in the 


accompanying diagram, the angle a meas- 
ures 40°, and the angle 6 35°. How many 
degrees does the angle ¢ contain? ‘The 


angled? The angle 1? The angle 2? 


4. Inscribe a regular nonagon in a circle of 2 inches radius, 
using the protractor. 


ELEMENTARY GEOMETRY. 591 


5. How many degrees does each angle of a regular nonagon 
contain? Draw a regular nonagon, each side measuring two 
inches. (Use the protractor.) 


6. The distance around a polygon is called its perimeter. 
What is the perimeter of a regular hexagon, inscribed in a circle 
whose radius is 1 inch? 

What is the circumference of the circle? 


7. The distance from the center of a regular polygon to the 
middle point of one side is called the apothem. 
Draw the apothem of a regular hexagon inscribed in a circle 
of 1 inch radius. About how long is it? 


8. Cut a regular hexagon, side one inch, into six triangles. 
Place three in a line, and fit in the other three so as to make a 
rectangle. (Divide one of the triangles into two equal parts.) 

How long is the base of the rectangle? What part of the 
perimeter? About how long is the perpendicular of the rect-_ 
angle ? | 


9. In a circle, radius 1 inch, inscribe a regular octagon. 
Divide it into eight triangles, and make out of them a rectangle. 

About what is the half perimeter of the octagon? Its 
apothem? Its area? 

Which has the greater perimeter, apothem, area, the hexagon 
or the octagon ? 


10. Find the approximate area of a regular hexagon, side 1 
inch, apothem about ¢ inch. 


11. Find the approximate area of a regular octagon, side about 
3 inch, apothem about +? inch. 


12. If we inscribe in a circle a regular polygon of 16 sides, 
will its perimeter be greater or less than that of the octagon? 
Which polygon will have the greater apothem ? 


13. If we inscribe polygons of 32, 64, 128, etc., sides, what 
will be the greatest perimeter we can have in a circle of 1 inch 
radius? What will be the greatest apothem ? 


592 ARITHMETIC. 


14. Draw a rectangle that will be about equal to a polygon of 
a million sides inscribed in a circle whose radius is 1 inch. 
Mark upon it the dimensions. Calculate the area, 


15. What is the area of a circle whose radius is 2 inches? 
16. Find the area of a circle whose diameter is 10 inches. 


17. Find the area of a circle whose circumference is 6.2832 
inches. 


18. Calculate the area of a sector of a circle whose radius is 
10 inches, the are of the sector being 60°. 


19. How many square inches are there between the circum- 
ferences of two concentric circles whose radii measure 3 and 
6 inches, respectively ? 


SURFACES OF SOLIDS. 


1275. Prisms, Cylinders, Pyramids, Cones. 


Notr.— The pupils should first examine a number of prisms and pyra- 
mids, right and oblique, regular and irregular, triangular, quadrangular, 
pentagonal, etc. Right and oblique cylinders and cones should also be at 
hand. 


A prism is a body bounded by plane faces, two of which are 
equal and parallel polygons, the remaining faces bemg paral- 
lelograms. 


A Corer Gq 
NEA NO ope 
D E K L 
B N M 


Figyiie Fie. oe Fia. 3. 


The two parallel faces of a prism are called its bases. The 
remaining faces taken together constitute its convex surface. 


ELEMENTARY GEOMETRY. 593 


In Fig. 1, ABC and DEF are the bases; in Fig. 2, the bases are GHIJ 
and KLMN; in Fig. 3} OPQRS and TUVWX. 
The sides AB, CH, etc., GH, LN, etc., QF, OT, etc., are called edges. 


1276. Prisms may be either right or oblique. The convex 
surface of a right prism consists of rectangles. 

Fig. 1 isa right prism; Fig. 2 is an oblique prism. 

Norre.— When a prism is spoken of, a right prism is meant unless the 
word oblique is used. 


The altitude of a prism is the perpendicular distance between 


the bases. 
AD, BF, or CE is the altitude in Fig. 1. GY is the altitude 
in Fig. 2. 


1277. The number of sides in each base determines the name, 
as triangular (Fig. 1), quadrangular (Fig. 2), pentagonal (Fig 
3), ete. 


A quadrangular prism whose 
bases are parallelograms is called 
a parallelopipedon. Fig. 4 is an 
oblique parallelopipedon. Fig. 5 
is aright parallelopipedon. Any 
two opposite facesofaparallelopip- / 
edon may be considered the bases. Fie. 4. Fig. 5, 


1278. When the bases are regular polygons, the prism is said 


to be regular. 


Fig. 1 is a right regular triangular prism; Fig. 2 is an oblique irregular 
quadrangular prism. 


1279. A cylinder is a body having two circular parallel 
plane faces, and one curved face. The plane faces 
are the bases. ‘The curved face 
Fia. 6. constitutes the convex surface. cay | 
Cylinders, like prisms, are either right or ay 
oblique. The altitude of a cylinder is the 
perpendicular distance between the bases. 


Fia, 7, 


594 ARITHMETIC, 


1280. A pyramid is a body whose 
convex surface is made up of triangles 
having a common vertex, the base of 
the pyramid being a polygon. 

Pyramids are either right or oblique; reg- Fic. 9, Fra. 10, 
ular or irregular; triangular, quadrangular, pentagonal, etc. 

In a right pyramid, each of the triangles that make up the 
convex surface, is isosceles. When, in 
addition, the pyramid is a regular one, 
these triangles will be equal to each 
other. 

The altitude of any of these equal tri- B 
angles constitutes the slant height of a 
right regular pyramid. The adltetude of 
a pyramid is measured by a line drawn C 
from the apex to the center of the base. 


A 


D 
Fre, 11, 


AG is the slant height of the square pyramid, Fig. 11. AF is its alti- 
tude. 


1281. The cone has a single 
circular base; its convex surface 
is curved, sloping to the apex. 
ct In the right cone, Fig. 12, HT is the 
--)y slant height, and HX is the altitude. 


ft-==L- 
| ates ’ 
ee LO is the altitude of the oblique cone, 


iT 


- ~ 


1282. Surfaces of Prism, Cylinder, Pyramid, Oone. 


1. Draw the developed convex surface of a square prism, 
height 8 inches, one side of base 1 inch. 


2. Draw the developed (entire) surface of a triangular -prism, 
height 3 inches, each side of base 2 inches. 


3. Draw the developed convex surface of a prism 3 inches 
high, each base being a triangle having sides of 1, 14, and 2 
inches, respectively. 


ELEMENTARY GEOMETRY. 595 


4. Find the convex surface of each prism, and show that the 
convex surface is found by multiplying the perimeter of the base 
by the altitude. 


5. Show that the convex surface of a cylinder is found by 
multiplying the circumference of the base by the altitude. 


6. Draw the developed convex surface of a square pyramid, 
slant height 3 inches, each side of base 2 inches. 
Cut it out of paper, leaving on one edge a small 
strip for gumming. Fold into a hollow pyramid, 
and measure its altitude. 


7. Show that the convex surface of a regular pyramid is 
found by multiplying the perimeter of the base by one-half the 
slant height. 


8. Make out of paper a hollow square pyramid whose alti- 
tude shall be 8 inches, each side of base 2 inches. 
A 

1283. The developed convex surface of a cone is a 
sector, ABDC. 

How many inches does the are BDC measure when the B 
diameter of the base of the cone is 2 inches? 

The slant height of the cone is the radius of the circle 
of which the sector is a part. AB, AD, or AC is the slant height. 


D 
& 
E 


9. Cut out of paper a semicircle, radius 3 inches, adding a 
narrow strip for gumming, and fold into a cone. 
What is the slant height of the cone? The diameter of the 
base? The radius of the base? The circumference of the base? 
What is the ratio between the radius of the base and the slant 
height? Between the diameter of the base and the slant height? 


10. Calculate the convex surface of the above cone, and show 
that it is equal to the circumference of the base multiplied by 
one-half the slant height. 


11. Find the diameter of the base of a cone made by folding 
a paper sector of 90 degrees (quadrant), the radius of the sector 
being 3 inches. What is its slant height? 


596 ARITHMETIC, 


When a sector of 60 degrees is used, what will be the diameter — 
and the slant height, the radius of the sector being 3 inches? 


(Make no allowance for overlapping.) 


12. If you wish to make a hollow paper cone whose slant 
height shall be 5 inches, and the diameter of whose base shall be 
3 inches, how many degrees should the arc of the sector contain? 


13. Draw the development of a right pyramid 4 inches in 
altitude, whose base is a rectangle 3 inches by 2 inches. Is the > 
altitude (slant height) of each of the four triangles the same? 


14. Calculate the slant height of each convex face of a rect- 
angular pyramid whose altitude is 12 inches, and whose base 
measures 10 inches by 18 inches. 

Find the convex surface. 


1284. Surface of Frustum of Pyramid and Cone, 


When the upper part of a pyramid or of a cone is cut off by 
a plane parallel to the base, the remaining part 


is called a frustum. Da 


15. Draw one face of the frustum of a square 
pyramid. Of the frustum of a triangular pyr- 
amid. What figure have you drawn? 
Calculate its surface when the length of the 
upper side is 4 inches, that of the lower side is 8 inches, and the 
slant height of the frustum is 10 inches. 


16. Draw the developed convex surface of the frustum of a 
regular triangular pyramid, each side of the upper base measur- 
ing 1 inch, of the lower base 2 inches, the slant height being 
inches. i 

Suaerstion. — Locate the apex of the whole pyramid of which the given 
frustum forms a part. 

17. Find the convex surface of the frustum of a square 
pyramid, one side of the upper base measuring 2 feet, of the 
lower base 3 feet, and having a slant height of 4 feet. 

Find the entire surface. 


ELEMENTARY GEOMETRY. 597 


18. Show that the convex surface of the frustum of a pyramid 
is equal to one-half the sum of the perimeters of the upper and 
the lower bases multiplied by the slant height. 

19. Draw the pattern of a small shade for a 
candle. Make the upper opening 14 inches in 
diameter, the lower one 24 inches in diameter, and 
the slant height 2 inches. 

20. How many square inches of tin will be required to make 
a pan, its upper base being 9 inches in diameter, 


the lower base 6 inches in diameter, and the slant 
height 4 inches? 
(Do not forget the bottom of the pan.) 


1285. The frustum of a cone may be considered the frustum 
of a pyramid whose bases contain a very great number of sides. 
The convex surface of the frustum of a cone may, therefore, be 
found by multiplying the half sum of the circumference of the 
two bases by the slant height. 

21. Find the convex surface of a frustum of a cone, the cir- 
cumferences of the bases being 15 inches and 20 inches, respec- 
tively, and the slant height 10 inches. 

22. How many square yards are there in the entire surface of 
a frustum of a cone, the radius of the upper base (7) being 3 
yards, of the lower base (4) 5 yards, and the slant height 6 
yards? | 

Circumference of upper base = 27r; of lower base = 27 R. 


Convex surface = (27r + 27R) x ee = t(r + &) Xslant height. 


Surface of upper base = mr?; of lower base = 7 R?2. 
Entire surface = 7 X what? Multiply only once by 3.1416. 
23. The diameter, AB, of the upper base of the E 


frustum of a cone measures 6 feet, CD measures bay 
8 feet, the slant height AC measures 9 feet. Find SO 
the slant height 4A of the part cut from the cone Las 
in making the frustum. 
Let HA=2; HC=24+9; AB=6; CD=8. 
The triangles HAB and ECD are similar. 


598 ARITHMETIC. 


24. Find the convex surface of the whole cone, HCD, and the 
convex surface of the part cut off, HAD. 


1286. The Sphere. 


A sphere is a body all points on 
whose surface are equally distant 
from the center. 

The distance from the center to 
the surface is called the radius 
of the sphere. The diameter is a 
line running between two points on 
the surface and passing through 
the center. 

CG, CK, CD, CF, CA, and Cl are radii; AD and FG are diameters. 


1287. If a sphere be cut through at any part, the cut surface 
will be a circle. When the cutting plane passes through the 
center of the sphere, the circle is called a great circle; other cir- 
cles are called small circles. 

FXGC is a great circle; HYIB and JLEZ are small circles. 


25. Find the length of an are of 60° of a great circle of a 
sphere whose circumference is 25,000 miles. 


26. Calling the are AJ in the preceding figure, 30°, the angle 
BCT will measure 30°. Calculate the radius BJ of the small 
circle when the radius CZ of the large circle is 4,000 miles. 
(LAA = are of O00) A == Ghordcon60.)) 


27. If Z is 60° from G, a point on the equator, find the length 
of the circumference of the small circle YJ, assuming the cir- 
cumference of a great circle to be 25,000 miles. 


28. What is the ratio between the length of a degree on the 
small circle YJ, and the length of a degree of a great circle? 


29. Calculate the radius of a small circle formed by passing a 
plane parallel to GCXF through a point on GA 45 degrees 
from GQ, 


ELEMENTARY GEOMETRY. 599 


1288. Surface of Sphere. 


We have seen (Art. 1151) that it may be experimentally shown 
that the surface of a sphere is equal to the 
surface of four of its great circles. 

Calling the radius of the sphere A, its 
surface 1s 4 ft’, 


\W 

4, 

aK 
yi 
4 ig 
Ay 

7 RR 


7 


30. Find the surface of a sphere whose 
diameter is 6 inches. 


31. How many square inches are there 
in the convex surface of a hemisphere whose radius is 3 inches? 

What is the area of the great circle that forms the base of the 
hemisphere ? 

Find the entire surface of the hemisphere. 


32. Is there any difference between the convex surface of a 
sphere and its entire surface? Why? 


VOLUMES. 
1289. Prisms and Cylinders, 


1. How many one-inch cubes will cover the base of a box 4 
inches by 8 inches? If the box is 2 inches deep, how many one- 
inch cubes will it contain? How many cubic 
inches are there in the volume of a right prism 
whose base is a rectangle measuring 4 inches by 
3 inches, and whose altitude is 6 inches? 


2. If the above hollow prism were divided 
into two equal parts by a thin partition extend- 
ing from a vertical edge to one diagonally oppo- 
site, how many cubic inches of sand would each 
part contain ? 


3. How many cubic inches are there in the 
volume of a prism whose base is a right-angled 
triangle 3 by 4 by 5 inches, and whose altitude 
is 6 inches? 


600 ARITHMETIC. 


4. Find the volume of a triangular prism, the area of the 
base being 6 square inches, and the altitude 6 inches. 

Find the volume of a triangular prism, each side “JS 
of whose base measures 6 inches, its altitude being 
8 inches. b 


5. What are the solid contents of a pentagonal 
prism formed by fastening together three triangular 
prisms whose bases contain, respectively, 12, 16, and 
18 square inches, the altitude of each being 15 inches? 


6. Ifa very great number of triangular prisms of 
the same height are united so as to form a cylinder 
whose base contains 12.5664 square inches, and whose altitude 
measures 5 inches, what are the solid contents? 


1290. Pyramids and Cones. 


With a center at C, and a convenient radius, describe an arc AB. Mark 
off four equal portions v, w, «, and y; and draw the equal chords. Cutting 
out CAvwey, with an additional narrow 


strip along Cy for gumming, and creasing C iy 
along the lines Cv, Cw, Cx, and Cy, we can 
fold the paper into a square pyramid. 

Measure its altitude, and make a square As 


_ prism of equal altitude and with an equal 
base. A 

Filling the pyramid with sand, and pour- -; os 
ing the sand into the prism, it will be found 
that the latter will contain the contents of the former three times; that is, 
the volume of a square pyramid is one-third that of a square prism having 
an equal base and an equal altitude. 

The same ratio will be found true in the case of a triangular, or any 
other pyramid, as compared with the corresponding prism, and of the cone 
as compared with a cylinder. 


1291. The volume of a pyramid or of a cone is equal, there- 
fore, to the area of the base multiplied by one-third of the 
altitude. 


ELEMENTARY GEOMETRY. 601 


1292. Frustums of Pyramids and Cones, 


7. Find the volume of a square pyramid whose altitude is 
12 inches, one side of the base measuring 6 inches. 
Find the volume of a square pyramid whose 
altitude is 6 inches, one side of the base measuring 
3 inches. 


8. Find the volume of the frustum of a square 
pyramid whose altitude is 6 inches, one side of the 
upper base 3 inches, and one side of the lower base 6 inches. 


9. A square pyramid whose altitude measures 18 inches, 
and each side of whose base measures EB 
15 inches, is divided into two parts by \ 
a plane, GHZ, parallel to the base, iT 
the distance, HJ, of the plane from 
the vertex, 4, being 6 inches. 

The ratio between the edge, #B, 
of the whole pyramid and the edge, 4 ay: 
HG, of the part cut off will be equal 
to that between HV and #X; that 
is, 6:18=1:3. The same will be B 
the ratio between BC and GH, and the latter will be one-third 
of 15 inches long, or 5 inches. 

Find the volume of the large pyramid and that of the small 
pyramid. 


\ 


| aK 


10. Each side of the upper base of the frus- 
tum of a square pyramid measures 5 inches; 
each side of the lower base measures 15 inches; 
the perpendicular distance between them meas- 
ures 12 inches. Find the solid contents. 

Find the convex surface of the above frustum. Find its 
entire surface. 


Note. — What is the slant height? 


602 ARITHMETIC. 


11. Find the total volume of three square pyramids, the alti- 
tude of each being 12 inches, and the areas of their bases being 25 
sq. in., 225 sq. in., and 75 sq. in., respectively. 


V75 15 5 


12. Find the number of cubic feet in a block of stone whose 
shape is that of a frustum of a square pyramid 4 feet high, each 
side of the upper base measuring 3 feet, and each side of the 
lower base 5 feet. 


1293. The volume of the frustum of a pyramid is equal to the sum of 
the volumes of three pyramids, each having an altitude equal to that of the 
frustum ; the base of one of them being equal in area to that of the lower 
base of the frustum, the base of a second being equal in area to that of the 
upper base of the frustum, and the base of a third being a mean propor- 
tional between the area of the other two. 

Base of first =3 x 3 sq. ft.; of second, 5x 58q. ft.; of third, V9 x 25 
sq. ft. = 15 sq. ft. 

Norr.— The mean proportional between two numbers is equal to the 
square root of their product. , 


13. Find the volume of the frustum of a square pyramid, its 
upper base containing 64 square inches, and its lower base 196 
square inches, its altitude being 18 inches. 


1294. Note that the mean proportional between 64 and 196 is 8 x 14, 
or 112. Since each is multiplied by one-third of the altitude, the opera- 
tion is shortened by adding together the three areas, 64, 196, and 112, and 
multiplying their sum by one-third of 18. 


Calling the altitude A, the side of the large square S, of small square s, 


the volume V, we have 
24 ) 


ELEMENTARY GEOMETRY. 603 


1295. The volume of the frustum of a cone is found in the same way 
as that of the frustum of a pyramid. 


+ altitude (area upper base + area lower base + area mean proportional). 


Calling the radius of the upper base r, and that of the lower base R, the 
area of the upper base will be mr’, of the lower base wA?, of mean propor- 


tional mr. 
=1A(mr? + wh? + ark) 


Since w (or 3.1416) is a common factor, we can save time by first adding 
r?, #?, and rf, and then multiplying by 7. 


ae: An(r? Le + rk) 


14. The diameters of the bases of the frustum of a cone meas- 
ure 8 and 15 inches, respectively; the altitude is 18 inches. 
Find the volume. 


15. How many cubic inches of water will a pan hold, whose 
lower base is 12 inches in diameter, whose upper base is 16 inches 
in diameter, and whose depth is 6 inches? How many gallons? 


1296. The pupils should make a frustum of a square pyra- 
mid of convenient size, and the three corresponding pyramids, as 
given in the rule. Fill the latter 
with sand, and pour the contents 
of all three into the frustum. 

To make the frustum, draw 
two concentric circles. Lay off 
equal arcs, AB, BD, DE, EF. 
Draw the chords and radii from the 
extremities of each chord. Draw 
the chords ab, bd, de, and éf. Cut 4 
out, after constructing a square fom 
either the upper or the lower base, 
and taking care to provide flaps 
for pasting. 


604 ARITHMETIC. 


To get a mean proportional between ab and AB for one side of 
the base of the third pyramid, lay off a line 7/7 equal in length to 
ab+ AB. On this line construct a semicircle. ee 
Make 1X equal to A, and at X erect a perpendic- 
ular KM. KM is a mean proportional between f ye 
ab and AB. 


1297. Oblique Prisms, 


We have seen that a rectangle, ABCD, and a parallelogram, — 
EFGH, are equal in area when the 


bases, AB and #F’ and the altitudes, i & 7a 7 
AD and HX, are equal, each to each. Vib 
This can be shown by cutting both 
A BE F 


out of paper, and by shifting the tri- 
angle HEX to the right side of the parallelogram. 


1298. In a somewhat similar way, we can show that an 
oblique prism is equal te a 
right prism that has an equal 
base and an equal altitude. 

Make from a potato or a tur- 
nip an oblique prism having 
rectangular bases, and change it to a right prism of the same 
height by cutting and shifting a portion. 


1299. The volume of any prism (or cylinder) is found by 
multiplying the area of the base by the altitude. 


1300. In the same way it can be shown that the volume of 
any pyramid or cone is equal to the product of the area of the 
base by one-third the altitude. 


1301. The Sphere, 


A sphere may be considered as made up of a great number of 
pyramids whose bases together make the surface of the sphere, 


ELEMENTARY GEOMETRY. 605 


and whose vertices all meet at the center of the sphere, making 
their altitudes each equal to the radius of the sphere. 
The volume of a sphere is equal, therefore, to its surface x 4 
radius. 
Surface = 4 great circles = 47h’; 
Volume = 47? x ‘ = tie 


16. Find the volume of a sphere whose radius is 9 inches. 


17. What is the volume of a sphere whose diameter is 9 
inches ? 

Find the volume of a cone whose altitude is 9 inches, diameter 
of base 9 inches. 

How does the volume of the cone compare with the volume of 
the sphere? 

How does the volume of the sphere compare with the volume 
of a cylinder 9 inches in diameter and 9 inches high ? 


1302. Take a clay sphere of a convenient size. Make a paper 
cylinder that will exactly contain it, the height of the cylinder 
being equal to the diameter of the sphere. Make a hollow cone 
of the same diameter and altitude. 

Place the sphere in the cylinder, carefully fill the cone with 
water, and pour it into the cylinder, which should then be filled 
to the top, showing that the volume of the cylinder is equal to 
that of the sphere and the cone together. 


APPENDIX. 


Se eeeneneaR a3 Gen Sa 


| TABLES. 


1303. Measures of Extension. 


Lona MEASURE. 


12 inches (in.). . . 1 foot (ft.) 
SPCCuL Aa pire 2 1 yard (yd.) 
Ber VEC Matera nthe 1 rod (rd.) 
SoU TOdeaet ie) 2h. 1 mile (mi.) 


A furlong (fur.) = 4 mile. 


Surveyors’ MEASURE. 


SPAR DCE A a ge 1 link (li.) 
BOLIN ke eka: 1 chain (ch.) 
SUA CEE Te. IEEE aR Ee 1 mile 


1 chain = 4 rods = 66 feet. 


SquaRE MEASURE. 


|e nS ee 1 square foot 
9 sq. ft. . l square yard 


304 sq. yd . . 1 square rod 
POUIS08 Pde mn oh 1 acre 
640 acres . . 1 square mile 


10 square chains = 1 acre. 


CuBic MEASURE. 


Di2e cunin. . .)., L.cubic foot 
Bide OTM Ges iic eo Bb 1 cubic yard 
lecord ~=128 cu. ft. 


1 bushel = 2150.4 cu. in. 
1 gallon = 231 cu. in. 


Measures of Capacity. 


Dry MEASURE. 


2 pints (pt.) . . . 1 quart (qt.) 
Siquarte mis, 1 peck (pk.) 
4 pecks.\. 4. 6. 1 bushel (bu.) 


Liquip MEASURE. 
2 pints (pt.) . . . 1 quart (qt.) 
4quarts .... 1 gallon (gal.) 

A gill is 4 pint. . 

The capacity of tanks, vats, etc., 
is frequently expressed in barrels of 
31% gallons. 

1 qt. dry measure =674 cu. in. 

1 qt. liquid measure = 57$ cu. in. 


Measures of Weight. 
Troy WEIGHT. 
24 grains (gr.) 1 penny weight 


(pwt.) 
20 pennyweights, 1 ounce (oz.) 


12 ounces 1 pound (1b.) 


Troy weight is used in weighing 
gold, silver, precious stones, etc. 


APOTHECARIES WEIGHT. 


20 grains (gr) . 1 scruple (Dd) 


dscruples ..... 1 dram (3) 
Sra nigu ets 1 ounce (3 ) 
12 ounces ... . 1 pound (fb) 


608 


Apothecaries’ weight is used in 
prescriptions. Drugs are bought and 
sold by avoirdupois weight. The 
grain, the ounce, and the pound 
apothecaries’ weight are the same 
as the corresponding denominations 
of troy weight. 


Avorirpupots WEIGHT. 


16 ounces (0z.) . 1 pound (1b.) 
2000 pounds. ... 1 ton (T.) 


1 lb. avoirdupois = 7000 grains. 
1 lb. troy eet LOU eee 


QM 1 (a3 


loz.avoirdupois= 4374 
1 oz. troy = 480 “ 


In weighing ores and coal at the 
mines and in calculating duties at 
the U. 8. custom houses, the follow- 
ing table is used: 

AO DOUDOS matt emer ue 1 quarter (qr.) 
4 quarters, 1 hundredweight (cwt.) 
20 hundredweight .... 1 ton (T.) 

1 ewt. = 112 lb.) TT) = 2240 Ib. 


Measures of Value. 


U.S. Money. 

DO trate ik ee 1 cent 
LOU CENTS fe hte ale 1 dollar 
1 dime = 10ce. 

1 eagle = $10. 


EneuisH Money. 
12 pence (d.) . . 1 shilling (s) 
20 shillings . 1 pound (£) 


1 farthing = + penny. 
£1 = $4.8665. 


ARITHMETIC, 


The Canadian dollar is equal 
in value to that of the United 
States, and is also divided into 
100 cents. 


The French frane (fr.) = 
19.3f, is divided into 100 cen- 
times (c.). 


The German mark (reichs- 
mark) (M., m.) = 28.8f, is di- 
vided into 100 pfennigs (pf.). 


Circular Measure. 


60 seconds ("’) . . 1 minute (') 
60 minutes... . 1 degree (°) 
360 degrees . 1 circumference 


Time Measure. 


60 seconds (sec.) 1 minute (min.) 


6Osminutes 2. ars 1 hour (hr.) 
24 HOUrSs sos es ste 1 day (da.) 
y Gaye wove ue ae 1 week (wk.) 
365 days 1 common year (yr.) 
BOO GAYS vu ae ee 1 leap year 


Years divisible by 4 and not 
by 100 are leap years. 


1892, 1896 are leap years. 


Years divisible by 100 but 
not by 400 are common years. 


1700, 1800, 1900 are common 
years; 1600 and 2000 are leap years. 


APPENDIX. 609 


1304. Time between Dates. 


1. In the common method, by compound subtraction, each 
month is considered one of 30 days, regardless of its length. 

EXAMPLE: Find the time between May 18, 1895 and March 
2, 1899. 


1899 3 2 Taking 30 days to the month, the 
1895 ee 8: difference in time is found to be 3 
3 Se ks) years, 9 months, 19 days. 


2. A more exact method is to take first the number of entire 
years, then the number of entire months, and lastly the number 
of days. 

May 13, 1895 to May 13, 1898 = 3 years. 

May 13, 1898 to Feb. 13, 1899 = 9 months. 

Feb. 13, 1899 to Mar. 2, 1899 =17 days. 
Ans. 3 years, 9 months, 17 days. 


8. Another method is to find the difference in years and days. 


May 13, 1895 to May 13, 1898 =3 years. 
May 13, 1898 to Mar. 2, 1899 = 293 days. 
Ans. 3 years, 293 days. 
1305. Days of Grace. 


Days of grace are not allowed in California, Connecticut, 
District of Columbia, Idaho, Illinois, Maryland, Massachusetts, 
Montana, New Jersey, New York, North Dakota, Ohio, Oregon, 
Pennsylvania, Utah, Vermont, and Wisconsin. 


Answers in which days of grace are not included, are inclosed in a 
parenthesis. 


When a note falls due on a Sunday or a legal holiday, it is 
generally payable on the next preceding business day. Some 
states, however, do not require payment until the next business 
day following. In these latter cases, banks include the extra 
days in calculating discount. 

In a majority of the states, days of grace are not allowed on 
sight drafts 


610 


1306. Rates of Interest. 


ARITHMETIC. 


The following table shows the rates of interest allowed by law 
in the several states and territories, the first column giving the 
rate allowed where none is specified in the note or other docu- 


ment. 


States and Territories. 


Alabama 4°). oy. 2 
AYIZON GE. eter ies 
ATE ONSRBi loys els Py 
Cig lifornia.:) ecco ne 
OLOTACOM, ca cee 
Connecticut 

Delaware...... 
District of Columbia 
Mlorida "0075 sae ns 
CLeOTgZiat) hy. sare 
TARNOD vee Ones ae 


Indiana. ©. .fer5e% 
Indian Territory. . 
LOWS ages cota ns is 
KANG RS We lieth ore 
Kentucky ..... 


Louisiana 


Maryland ..... 
Massachusetts .. . 
Michigan.) J 4/4 
Minnesota ..... 


Mississippi ..... 
Missouri 


© ©. @ (© ‘¢. 


Legal 
Rate. 


Contract 
Rate. 


Any rate up to the one given in the second column is 
permitted when an agreement is made in writing. 


Contract 
Rate. 


I 0 OD @D DD © ~T O =~T Co 


4 
a) 


Cou So) J Cd) (Od) Od -O>. Cr Od9 Od Cn) Od Op. Ct 


Montanay sien A 
Nebraska ..... 
Nevada. sks: 
New Hampshire . 


New Jersey... . 
New Mexico ... 
New York i 
North Carolina. . 
North Dakota 


Orevon a ase tas 
Pennsylvania 

Rhode Island. . . 
South Carolina. . 


South Dakota 


Virgenes eae 
Washington : 
West Virginia . . 
Wisconsin. .... 


Wyoming. ope..ss 


10 


DAODA~ATAPHHAHAAWAAANTNAI AAAI a nsw oOOoganrinona -t 


APPENDIX. 611 


PARTIAL PAYMENTS. 


1307. Connecticut Rule. 

Find the amount of the principal to the time when the payment, 
or the sum of two or more payments equals or exceeds the interest, 
uf such time rs one year or more from the tume the interest com- 
menced. From this amount deduct the payment or sum of 
payments. Use the balance then due as a new principal, and 
proceed as before. 

Tf, however, any payments exceeding the interest then due, are 
made within a year from the time from which the interest is 
reckoned, the amount of the principal is calculated for one year, 
and from this amount are deducted the amounts of such payments 
from the tueme made until the end of the year. When the settle- 
ment 1s made in less than one year from the tume from which 
anterest 1s reckoned, the amount of the principal and those of the 
payments are calculated to the date of settlement. 


5 Ae Norwicu, Conn., Jan. 5, 1896. 


Two years after date, we, jointly and severally, promise to pay 
to Emerson W. Keyes, or order, Three Hundred Dollars, value 


received, with interest at 6 per cent. JostaH H. Pitts, 
$ 300,29. CHARLES W. FIELD. 


Endorsements: May 20, 1896, $100; Nov. 2, 1896, $100; 
March 17, 1897, $50. 

How much is due Jan. 5, 1898? 
Amount of $300 to Jan. 5, 1897, l year . . . $ 318.00 


Amount of $100 from May 20, 1896 to Jan. 5, 1897, Thy mo. $103.75 
Amount of $100 from Nov. 2, 1896 to Jan. 5, 1897, 2 


A OeutaClegh RA 3h) Sie ean eamog uke AOL OD oun a anes 
Balance Jan. 5, 1897 Boe 4, Pea dim 9 Meat eS $113.20 
Interest on $113.20 to Jan. 5,1898.. . . . .... 6.79 
Amount.) > . $119.99 
Amount of $50 from March 17, 1897 Be ee 5, "1898, 
OVO G JIGME DTE oe, A eh aes 52.40 


DSR ANPOGLOOG ES 5 cuba) Sool a ao isl eh $67.59 


612 ARITHMETIC. 


2. DANIELSONVILLE, Conn., April 15, 1896. 

On demand, I promise to pay to the order of Robert P. Webb, 
Two Thousand Dollars, value received, with interest. 

$ 2000,%%. JoHN J. BARNICLE. 


On this note are endorsed the following payments: $50, 
Sept. 20, 1896; $100, May 26, 1897; $1,000, June 20, 1898. 
How much is due Dec. 27, 1899? 


Face of note . . » ve 2000.00: 
As the first aphettte $50, is nor ieee aecneh 3 nee the in- 
terest then due, the interest on $2,000 is calculated to date 


of second payment, May 26, 1897, lyr. lmo.llda. . . 133.67 
Amount die/May: 26, 18970 SI LP Re ae Sree 

Less payments, $o0 and $1009.) 25) heyaemey een 150.00 

Balance May 26,1897. . . . pul other tes UAB LESSis A 

Interest on $ 1,983.67 to June 20, 1898, 1 yr. 24 ie ss stele Wiad 126.95 
Amountidue Jane'20,11898) 4). salu aa) Same 

Less third payment). <iyar ad bets \ wh kek ce Unee ce me 

Balance June 20,1898 .. . pees te eh he 2 

Interest on $1110.62 to Dec. 27, 1899, 1 yr. 61 mo. 7, aan Mii 2 101.25 
Due Deo, 27,1899 UP ee, VO ts ee eee 


3. Find the amount due April 16, 1898, on an interest-bearing 
note for $1,000 drawn June 1, 1896; on which were paid $150, 
Sept. 16, 1896; and $50, Sept. 16, 1897. 

4, New Haven, Cory. , July 25, 1896. 

On demand, I promise to pay to the order of Frank Regan, 
Jr., Five Hundred Dollars, with interest at 6%. 

$ 500200. JAMES W. NAvGHTON. 

Endorsements: Sept. 18, 1896, $100; Feb. 5, 1897, $200. 

How much is due April 1, 1897? 

5. by aes Peas, HarrtrorD, Conn., Jan. 2, 1894. 

Three years after date, we promise to pay to a order of 
George G. Brown, Eight Hundred Seventy and 35% Dollars, 


value received, with interest. 
$ 870355. WuHittock, Tuuuy, & Co. 


APPENDIX, 


613 


Find the sum due at maturity on the above note, the following 
payments having been made: Sept. 12, 1895, $35; March 18, 


1896, $200; Aug. 24, 1896, $250. 


ANNUAL INTEREST. 


1308. Partial Payments. 


When payments have been made on a note subject to “ 


annual 


interest,” the amount due at any time is calculated as follows: 


Find the amount of the principal by the method gwen for 
annual interest (Art. 1172) until the end of the first year in which 
the amount of the payment or payments equals the interest then 
due. From the amount of the principal, subtract the amounts of 


the payments. 
proceed as before. 


Use the balance then due asa new principal, and 


6. Lansine, Micu., June 1, 1895. 
For value received, I promise to pay to the order of Jere. 
Smith, Six Hundred Dollars, with interest annually at six per 


cent. 


$ 600,95 


FRANK H. Cooper. 


Endorsed: Feb 1, 1896, $30; July 1, 1896, $100; Oct. 1, 


1897, $200. 
How much is due Oct. 1, 1898? 
Principal . 
Interest to June 1, 1897, 2y years . 
Interest on $36 wield interest, 1 year 
Amount June 1, 1897 , 
Amount of $30 Feb. 1, 1896 to June 1, 1897 


Amount of $100 July 1, 1896 to June 1, 1897 . 


New principal, June'1, 1897 °. 

Interest to June 1, 1898 

Amount June 1, 1898 : 
Amount of $ 200 Oct. 1, 1897 to June 1, 1898 

New principal June ‘1, 1898 

Interest to Oct. 1, 1898 

Amount due Oct. 1, 1898 . 


$ 600.00 
72.00 
2.16 

& 674.16 


137.90 
$536.26 
32.18 
$568.44 
208.00 
$ 360.44 
7.21 
$367.65 


614 ARITHMETIC. 


Y. Hiram Stevens, of Hallowell, Me., borrowed of John Karst, 
June 15, 1895, seven hundred dollars, agreeing to repay it on 
demand with six per cent per annum, and six per cent on the 
deferred payments of annual interest. The following payments 
were made: Nov. 15, 1895,$20; Feb. 15, 1897, $80; Sept. 15, 
1898, $15. Find the amount due Oct. 15, 1899. 


1309. New Hampshire Rule. 
In New Hampshire, partial payments on a note “ with interest 
annually ” are subject to the following rule: 


Tf the payments during a year equal or exceed the interest due 
at the end of the year, the amount of the payments to that date is 
deducted from the amount of the principal. 

If the payments are less than the wterest due, the amount 
of these payments is used (1) to cancel the interest due on the 
annual interest, and (2) to reduce the annual interest then due. 
But uf no interest 1s due except what is accruing during the year, 
only the principal of the payments is thus applied, no interest 
being added. 

Notr.— When a payment is less than the interest due on the annual 
interest, the remaining simple interest is not subject to interest. 

8. PortsmouTH, N.H., June 1, 1895. 

On demand I promise to pay to the order of W. M. Davis & 
Co., Six Hundred and ,%%, Dollars, value received, with interest 
annually, at six per ons 

P6003. Henry 0. McLean. 


Endorsements: Feb. 1, 1896, $30; July 1, 1896, $100; 
Oct. 1, 1897, $200. How much is due Oct. 1, 1898 ? 


PRN eIp a Ce hoe hatte tol Behe hehy epee aoa TOLL m 
Annual interest ave I une i! 1896 Sel is 2) hept auth Baca ice Sash Ea RCRD 
Payment Feb. 1,1896 . .. . of {ol eal O'pt to) Mma 
Balance of interest due June 1,1896 . ...... $6.00 
Annual interest due June 1,1897 ........ £96.00 
Interest on $6 forl year . . . oe eee) Bie eta de ae 36 42.36 


Amount due June 1; 1897.) erie $642.36 


APPENDIX. 615 


Brought forward . . . alten $642.36 

Amount of $100 July 1, 1896 to Fane i 1397 ii ad ta 105.50 
New principal’ Jane 118978. a $ 536.86 

Puterest on  036.86.to June: 1.1898 9. 2. i 32.21 
Amount aueld ite ESOS A ivw ee Wo) es ktes $569.07 
Amount of $200 Oct. 1, 1897 to June 11,1898 . ... 208.00 
New, principal June], 1808) ss. $ 361.07 

Interest on $361.07 to Oct..1, 1898. . 3... wt. 7.22 
POO Ob PN Lateral me mee loteb sar pauiae Wis) Dds ah $ 368.29 

9. Concorp, N.H., May 1, 1896. 


I promise to pay Albert 8. Caswell, on demand, Two Thousand 
Five Hundred Dollars, with interest annually. 


$ 250052. WALTER B. GoopNnouau. 


Endorsements : Oct. 1,1896,$100; June 1, 1897,$ 1,000; Nov. 
1, 1898, $50. Find the amount due Oct. 1, 1899. 


10. How much is due March 11, 1899, upon a note for $ 1,000, 
with annual interest, drawn Jan. 8, 1895, on which the follow- 
ing payments have been made: June 1, 1895, $10; March 14, 
1896, $10; Sept. 30, 1898, $500? 


Notr.— Ordinary interest-bearing notes on which partial payments 
have been made, are subject to the U.S. Rule. 


1310. Vermont Rule. 


Partial payments on “ annual interest ” notes are used in Ver- 
mont (1) to cancel the interest due on the annual interest; (2) to 
reduce the annual interest then due; (8) to reduce the principal. 


RULE. 


Tf the payments during a year equal or exceed the interest due 
at the end of the year, the total of the amounts of the payments is 
deducted from the amount of the principal. 

Tf the total payments are less than the interest due, the amounts 
of these payments are used (1) to cancel the interest due on the 
annual interest, and (2) to reduce the annual interest then due. 


616 ARITHMETIC. 


Norr. — When the amount of a payment is not sufficient to cancel all the 
interest due on the annual interest, the balance of this simple interest is not 
subject to interest. 

a1, Buruineton, Vt., June 1, 1895. 

Due Halsey Clark, Six Hundred Dollars, payable on demand, 
with interest at six per cent annually. 

$ 600555. CHARLES HERMAN. 


Endorsements: Feb. 1, 1896, $30; July 1, 1896, $100; Oct. 
1, 1897, $200. How much is due Oct. 1, 1898? 


la gtiternare yeh va ke Oeeuneakar Reabemer $ 600.00 


Annual interest ine pee 1, 1896 Ce Se eS eRe OD ee 
Amount of $30 Feb. 1, 1896 to June 11,1896 . . . . 30.60 
Balance of interest due June 1,1896 . . .... . $5.40 
Annnal intereshidue une 1807. eee ee eee 36.00 
liiterest' on. $5.40 \for'one year. Fase) ae 2 41.72 
Amount due June 1,1897... . penne $ 641.72 
Amount of $100 July 1, 1896 to June 1, 1897 . Saale. 105.50 
Now principal. Jone d, 1897 tore ee $536.22 
interest.on' $550.22 16.0 nte.. looceme ee ee Ce S2.Le 
Amount due June: 1 15980 ve ie meee ee $ 568.39 
Amount of $ 200 Oct. 1, 1897 to June 1,1898 . . . . 208.00 
New principal J une Td 898 ay erleia aera ne $ 360.39 
Interest‘on $360.39 to Oct, 1d; 1398F BC ay ae iene 7.21 
ne Oe Tega re eae een Ou oe _ $367.60 
12. RutTLAND, VT., June 1, 1895. 


I promise to pay Ahern & Bro., on demand, Two Thousand 
Five Hundred Dollars, with interest annually. 


$ 25005%%. Maenus ScHULER. 


Endorsements: Oct. 1, 1896, $100; June 1, 1897, $1,000; 
Nov. 1, 1898, $50. Find the amount due Oct. 1, 1899. 


13. Find the amount due March 11, 1899, upon a note for 
$1,000, with annual interest, drawn Jan. 3, 1895, on which the 


following payments have been made: June 1, 1895, $10; 
March 14, 1896, $10; Sept. 30, 1898, $500. 


APPENDIX. 617 


14. Brennineton, Vt., March 17, 1896. 

On demand, I promise to pay Francis de Raismes, Three 
Thousand Dollars, with interest annually. 

$ 3000-%0,. Epwarp HurtcHinson. 

Endorsements: May 17, 1899, $20; Sept. 17, 1902, $1000. 
How much is due March 17, 1903? 


15. A demand note for $1200, with annual interest, dated 
Feb. 25, 1893, bears the following endorsements: Feb. 26, 1896, 
$400; June 11, 1898, $10; Sept. 26, 1900, $400. What sum 
will pay the note in full Feb. 26, 1902? 


TAXES. 


1311. The real estate of a town in which a tax of $3,300 is to 
be raised, is assessed at $180,000, and the personal property at 
$20,000. There are 100 taxable polls at $2 each. What is 
F. D. Clarke’s tax, his real and personal property being assessed 
at $10,000, and who pays two poll taxes? 

$2 x 100 = $200, amount to be raised by poll taxes. 
$ 3,300 — $ 200 = $3,100, amount to be raised on property. 
$ 180,000 + $ 20,000 = $ 200,000, assessed value of property. 

$ 3,100 + 200,000 =153 mills, tax on $1, or the rate. 


$ 10,000 x .0155 =+=$155, tax on Mr. Clarke’s property. 
$2x 2 = $4, Mr. Clarke’s poll taxes. 
$155 + $4 = $159, Mr. Clarke’s total taxes. 


1. Find the taxes of W. E. Pulsifer, of the above town, whose 
real property is assessed at $8,500, whose personal property is 
assessed at $ 600, and who pays one poll tax. 

2. What is the rate of taxation in a town that proposes to 


raise $2,500, the real and personal property being valued at 
$ 275,000 and the number of polls at $2 each being 150? 


1312. Vermont Method of Levying Taxes. 
In Vermont, all male inhabitants between 21 and 70, unless 


specially exempted, are “listed”’ for poll taxes in their respective 
towns, at $2 each. 


618 ARITHMETIC. 


Real and personal property is placed in the town “list”’ at 
1% of its appraised value. 

The “ grand list’ comprises the total of the poll list and 1% 
of the assessed (appraised) value of the real and the personal 
property, and is the amount on which all taxes are levied. 


3. The real estate of a town in which a tax of $3,300 is to be 
raised, is appraised at $180,000, and the personal property at 
$20,000. There are 100 taxable polls. What is John Griffin's 
tax, who pays two polls, the appraised value of his real property 
being $7,500 and of his personal property, $2,500? 


$180,000 + $ 20,000 = $200,000, appraised value of real 
and personal property. 

rhs of $200,000 = $ 2,000 = 1% of appraised value. 

$2 x 100 = $ 200, the poll list. 

$ 2,000 + $ 200 = $ 2,200, the grand list. 

$ 3,300 + 2,200 = $1.50, rate on $1 of the grand list. 

[1% of ($ 7,500 + $ 2,500)] + ($2 x 2) = $104, John Griffin’s grand list. 

$ 1.50 x 104 = $156, John Griffin’s town tax. 


4. Find the city tax of Parker P. Simmons, whose property, 
real and personal, is assessed at $9,500, and who pays one poll, 
the rate being $2.45. 


5. How many taxable polls are there in a town in which the 
appraised value of the real and personal property is $ 150, 000, 
the tax rate necessary to raise $3,600 being $2? 

Let x = the number of polls; 2 “= poll list. 

6. Henry W. Hallock’s taxes on real and personal property, 
appraised at $12,000, amount to $252, including 3 poll taxes. 
The total amount levied is $6,800. Find the appraised value of 
all the property, the number of taxable polls being 200. 


1313. Values of Foreign Coins. 


The following estimate by the Director of the Mint, of the ~ 
values of foreign coins, has been proclaimed by the Secretary 
of the Treasury, January 1, 1897: 


APPENDIX. 619 


VALUES OF FOREIGN COINS. 


ars 
=I) 
& 8p 
~ wD , 
CounTRY. Standard. Monetary unit. mS K 
Se, '5 
3 os 
> 
Argentine Republic........... Gold and silver..... PORGie an deeiaiel ene $0.96,5 
Austria-Hungary ............. (CGR Seek eiche aeentor Crowng ts acanekl steals .20,38 
1 Folin) CASO amo bnie echoes e Gold and silver ..... CAT Cesta ae rsye ees icicle eet ste 19,3 
isXbhwhy AA seus ee os dete Shela SII? oie a eiierentn wine's BONViIanO) 26ers estas « 47,4 
Brazilyacia « <fefate tis leiise ts carats CXKNGUA tev onondece one IMIG 18s aera eieisers ccleon okt .54,6 
British Possessions N. A. (ex- 
cept Newfoundland) ...... (EOI Wioeinioobean conde Dollars. vs eesmasts skiers: 1.00,0 
Central Amer. States......... Silver ccaseee couse IPGSOU Riatdctccle side on! share «re 47,4 
(MBG socanbancces ponneneSmBer CPOl dete eee cietsis eran PESO/oastrs cipatita cite gen .86,5 
EATINIOY weraetere 76,7 
Cantons ..: 76,5 
Chetoo.> 5.12 .13,3 
Chin Kiang. 74,9 
Fuchau .... 70,9 
Haikwan... .78,0 
CHING Ree recess sie uese scien se Silveracsauseds acces Tele. ase Hankow .. ALT 
Niuchwang. CTS 
Ningpo .. BES 
Shanghai... .70,0 
Swatow .... .70,8 
Takanlesce .. ee 
Tientsin 74,8 
GWolombia vende sssoneeeee wees Silvera PSG ier OR ee ae ok AT,4 
Cubase ee eco s aacesee Gold and silver..... POO nSeos ee e eee eee 92,6 
Grab kertatcsrete sare. cccicle de sien cores OL Scott cee baa cle Crowle inion coke “ .26,8 
UCTIBCLOL setae a4 creme e's, vie eca selene Silvers. wen ee arrc ces PSULCh Gn trees Alena eine: 47,4 
SUT Phe cis oh apelin en's wSideriees Golde aera scweeeee s Pound (100 piasters)....} 4,94,8 
TTA CHAG. a5 AAR OSORIO EO BORe er Goldiarwecce on ate Mia rkterc ss sites td. loetter oe os 19,3 
JOT NS ct PRO ee SAE cm ier Gold and silver ..... LNs COB MOREL On OCE 19,3 
German Empire .............. Golde Rnewee sedescs IM air Keverne ste eisharet nevarane Dobe .28,8 
Speer) ayy UE aad Sodciemapo Hpccig GrOLGUAE Ne Sake cieve crs 8s Pound sietlingy\ vo. a0. <- 4,86, 64 
GCTECCER LE ene ia eee we vical Cold and silversseee| Drachinay -cesicee en ekuee 19,3 
URE Meyer v's eu-s acis ale crelsieraietacate Gold and silver..... GouUurdeye eee ee eee 96,5 
LGN og de pe SoA OEL a eae OEIee Silverio hase RU POC! aisionscearaieny owes 122.5 
Wfslivttdctistreceis ste ssita uae ascigs Gold and silver..... Drache atc eee tele 19,3 
MODAN Geass ns attra vest siess Gold and silver..... cons, eens Hants ar 
Liberia. ......-.......--2..--. reli fe RSP aces Wage tr DGUSE Ay rican ncet eens 1.00,0 
INSGXIGO Bere e tne cierto ee oimicote.s Silvas soem coerce Dolareeeeces vce FOES 
INGCHErIANUS se tease oe eos Gold and silver..... IW fey bia aan aoreees or .40,2 
Newfoundland................ ra aA ea Wee Miele ota eae cee gies 1.01,4 
INOUWHVamere «ccuwialat scien ie croee Gold CO Wises taco estas ae .26,8 
ROrsiaerrieyen oe aes c cules cca Silver sesame teams FTA re certs iste eres stein ers 08,7 
POF... 2... e ee ee cece e cece cnes Silverds sy. soeeccee : Boley wescantesre es 47,4 
Portugal .........-0sssceesees Goldivanaeco eae Milreig.€. calc ciate ts 1.08,0 
ELURSIB Bese wierceies oe ieee ote cacs Sil verve sass tne Ruble.... : ae ton en 310 
Fee OE Deli Har: es AER re ape AAR ae Gold and silvers.c.e nbesetanec sts iyeriefee « 19,3 
BW OU Git nua ttre Go teenth en ccc Golder ors net CEOW iets ce sean keels ke .26,8 
Pwitzenlandeoce voce seco nes Gold and silver..... Han Cacitee bart cio crateloa areas 19,3 
PUUEKOyieerseth stlathine die elena GOId. aaue wins eaters’ RoW) gan Sno oracle se 04,4 
MOTUS AY oe ales sla eam atde ed no nic Goldie en Sameera POSO eer aac tees oe 1.03,4 
WWONOZMO Al utas seus ole tore castarcat Gold and silver..... olivaree cme aescate 19,3 


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INDEX. 


The Roman numbers refer to pages in the Supplement. 


Abstract numbers, i. 
Acceptance, 451, xxvi. 
Accounts (see Manual), xii. 
Acurate interest, 494, 495. 
Definitions, etc., 494, xviii. 
- Acute-angle, 865, 557, 559, 561, 
565. 
Acute-angled triangle, 365, 
570, 571, 584, 585. 
Altitude of, 584, 585. 
Area of, 411, 412, 427, 584, 585. 
Construction of, 865, 565, 584. 
Addends, ii. 
Addition, of algebraic quantities, 


565, 


530, 531. 
Of decimals, 241, 283, 292, 440, 
441, 482. 
Rule, ix. 
Of denominate numbers, 236, 
270, 285, 287, 312, 329, 330, 
336, 858, 406, 430, 433, 449, 


484. 
Of fractions, 199, 200, 201, 202, 
208, 204, 205, 249, 255, 262, 
277, 309, 362, 364, 482, 433, 
483, 484. 
Principle and rule, 203, ix. 
Of integers, 211, 212, 217, 275, 
302, 339, 429, 430, 510. 
Cross-adding, 211, 216, 275, 
302, 510. 
Definition, principle, and rule, 
ii. 
Adjacent angles, 365, 557, 560, 
561. 
Ad valorem duty, xvi. 
Algebraic equations, one unknown 
quantity, 869-380, 534-538, 
550-556, 


Algebraic equations, Continued. 
Clearing of fractions, 373. 
Transposing, 377. 

Removing parentheses, 534. 

Quadratics, affected, 551-556. 
Pure, 549-551. 

Three unknown quantities, 544- 
547. 

Two unknown quantities, 588- 
544. 

Algebraic quantities, addition of, 
530, 531. 

Multiplication of, 534, 547, 548, 
549. 

Square root of, 552, 553. 
Subtraction of, 532, 533, 584. 
Aliquot parts, U. S. money, 218, 
219, 220, 221, 222, 228, 224, 
225, 272, 273, 803, 804, 335, 
397, 427, 447, 455, 475, 507. 

Interest by, 417, 418, 419, 420, 
469, 470, xvii. 

Miscellaneous, 399, 405, 410, 421, 
422, 429, 455, 471, 474, 484. 

Alligation, 502, 5438, 544. 

Altitude, of parallelogram, 366, 
412, 569, 570, 578, 584. 

Of solid, 511, 593, 594, 595, 596, 
604, 

Of triangle, 367, 412, 495, 496, 
504, 508, 556, 569, 578, 584, 
585. 

Amount, addition, ii. 

Interest, 394, xvi. 

Percentage, xiii. 

Analysis (see Cause and effect), 
230, 236, 258, 259, 263, 264, 
265,270, 271, 299, 300, 3is, 
318, 829, 336, 352, 353, 362, 


621 


622 TNDE Xe 


Analysis, Continued. 
364, 407, 409, 423, 436, 437, 
438, 489, 442, 454, 461, 462, 
477, 478, 479, 482, 483, 492, 
493, 509, 510. 

Angles, acute, 865, 557, 559, 561, 
565. 

Adjacent, 365, 557, 560, 561. 

Bisection of, 577. 

Construction of, 559, 560, 561, 
562, 563, 564, 565, 566, 567, 
569, 577, 578, 579, 580, 581, 
582, 585. 

Designation of, 558. 

Exterior, 565. 

Formed by parallel lines and a 
secant, 563, 564, 565, 590. 

Kinds of, 365, 557. 

Measurement of, 558, 559, 560, 
561, 562, 563, 564, 565, 566, 
567, 569, 577, 578, 579, 580. 

Oblique, 557. 

Obtuse, 365, 557, 561, 566. 

Of parallelogram, 569, 590. 

Of polygon, 574, 579. 

Of triangle, 564, 565, 579. 

Opposite, 557, 561. 

Right, 565, 491, 557, 559, 560, 
561. 

Vertical, 557, 561. 

Annual interest notes, 523, 613- 
617, xviii. 

Partial payments on, 613, 614. 
New Hampshire rule, 614, 615. 
Vermont rule, 615, 616, 617. 

Antecedent, 466, xxi. 
Apothecaries’ weight, 361, 406, 607. 
Apothem, of regular hexagon, 591. 

Of regular octagon, 591. 

Applications, of algebra, 383, 384, 
385, 387, 389, 400, 409, 415, 
422, 424, 434, 452, 460, 480, 
490, 498, 502, 503, 555. 

Of percentage (see Percentage). 

Of square root, 270, 485, 486, 
490, 491, 508, 504. 

Approximations, 226, 282, 305, 306, 
353, 360, 361, 398, 442. 
Arabie notation, 210, 240, i. 
Arc, 516, 572, 577, 580. 
Bisection of, 576, 577. 


Are, Continued. 

Length of, 450, 451, 572, 
580, 590, 598. 

Area (see Measurements). 

Of circle, 461, 496, 503, 504, 516, 
517, 526, 592. 

Of rectangle, 246, 247, 248, 262, 
276, 283, 284, 286, 288, 289, 
290, 297, 800, 301, 315, 338, 
342, 343, 344, 345, 352, 353, 
054, 855, 356, 389, 390, 397, 
398, 400, 401, 411, 412, 413, 
434, 435, 486, 461, 462, 475, 
476, 482, 504, 506, 517, 525, 
526, 550, 555, 584. 

Of regular polygon, 591. 

Of hexagon, 504, 591. 

Of octagon, 591. 

Of rhomboid and rhombus, 367, 
412, 413, 496, 508, 504, 570. 

Of ring, 517, 592. 

Of sector, 516, 592. 

Of segment, 516. 

Of trapezium, 368, 413, 495, 504, 
508, 571. 

Of trapezoid, 367, 368, 412, 413, 
434, 490, 491, 504, 571. 

Of triangle, oblique-angled, 367, 
411, 412, 461, 495, 496, 503. 

Right-angled, 322, 823, 355, 
356, 367, 411, 412, 453, 490, 
496, 526, 556. 
Assessed value (see Taxes), 385, 
407, 444, 446,510, 617, 618, xvi. 
Assets, xxii. 
Avails (see Proceeds), 402. 
Average, of accounts, 500. 

Term of credit, 499, 500, 501. 

Rules, etc., xxi. 

Avoirdupois weight, table, 227, 
361, 406, 608. 
Long ton, 541, 608. 


577, 


Balance, xii. 
Bank check (see Manual), xxvi. 
Bank discount, definitions, etc., 
402, 403, xix. 
Of interest-bearing notes, 407, 
408. 
Of notes without interest, 402, 
403, 404, 420, 421, 455, 


INDEX. 


Bank discount, Continued. 

Proceeds (avails), 402, 403, 404, 
420, 421, 483. 

Term of discount, 402, 403, 404, 
421, 424, 470, xix. 

To find face of note, etc., 424, 
425, 470, 494. 

Barrel, 607. 

Base, bonds, 444, 488, xxv. 
Commercial discount, xiv. 
Commission, 444, xv. 

Duties, xvi. 

Insurance, 444, xv. 

Line (U.S. lands), 524. 

Of isosceles triangle, 366, 567. 
Of prism, 592. 

Percentage, 3838, xiii. 

Profit and loss, 386, 387, xiv. 
Stocks, 444, 448, xxv. 

Taxes, 444, xvi. 

Bill, 288, 239, 274, 319. 

Definitions, etc., xii. 

How receipted, xii. 

Of exchange, 448, 455, 456, 457, 
XXvVl. 

Bins and tanks, capacity of, 359, 
560, 361, 389, 402, 406, 433, 
435, 442, 461, 482, 514, 527. 

Bisection of angle, 577. 

Of arc, chord, and sector, 576, 
yee 
Of line, 575. 

Board, foot, 514, 515, 516. 
Measure, 515. 

Bonds and stocks, 888, 442, 443, 

444, 445, 488, 489. 
Brokerage, 4438, 444, 488. 
Definitions, etc., 442, xxv. 

Broker, xxv. 

Brokerage, 443, 444, 488, xxv. 

Business forms, 238, 274, 319, 394, 
402, 407, 448, 451, 455, 458, 
460, 466, 496, 498, 523, 613, 
614, 615, 616, 617. 

Definitions, xxiv. 


Calculation of heights and dis- 
tances, 436, 587, 588, 589, 590. 

Canadian money, 608. 

Cancellation (see Manual), 263, 
264, 265, 275, 289, 295, 3807, 


623 


Cancellation, Continued. 
318, 314, 850, 862, 889, 3890, 
402, 414, 435, 487. 
Definitions, etc., 268, 264, v. 
Capacity of bins and tanks, 359, 
360, 361, 389, 402, 406, 483, 
435, 442, 461, 482, 514, 527. 
Capital, xxii. 
Capital stock, xxiv. 
Carpeting, 301, 3845, 889, 401, 434, 
461, 462, 492. 
Cause and effect (see Analysis), 
436. 
Chain, surveyors’ 
461, 491, 607. 
Check (see Manual), xxv. 
Chord, 572. 
Bisection of, 576, 577. 
Length of, 516, 572, 582, 590, 
598. 
Circles, area of, 461, 496, 508, 504, 
516, 517, 526, 592. 
Circumference of, 450, 468, 469, 
503, 571. 
Circumscribed, 504, 582. 
Concentric, 572, 592. 
Diameter of, 450, 468, 469, 503, 
571, 592. 
Great, 518, 522, 598. 
Inscribed, 582. 
Radius of, 469, 496, 503, 571. 
Small, 598. 
Tangent, 572, 584. 
Circular measure, 450, 558, 608. 
Circumference, 450, 468, 469, 508, 
571. 
Circumscribed, circle, 504, 582. 
Polygon, 573, 574, 577, 582, 583, 
584. 
Square, 574, 577. 
Triangle, 574. 
Clearing of fractions, 373, 477. 
Coefficient (see Manual), 530. 
Commercial discount, 410, 411, 
421, 422, 431, 446, 456, 457, 
475, 479, 480, 481. 
Definitions, xiv. 
Commission, 317, 384, 385, 398, 
423, 449, 482. 
Agent, xiv. 
Definitions, ete., xiv. 


measure, 441, 


€ 


Common, denominator, 206, 267, 
284, ix. 

Factor, 252, v. 

Fraction, vii. 

Multiple, vi. 

Prime factor, v. 

Complex, decimal, 292, 296, x. 

Fraction, viii. 

Reduction of, 266, 268, 271, 
324, 364, 4338, 484, x. 

Composite, factor, v. 

Number, v. 

Compound, addition, 286, 270, 285, 
287, 312, 829, 330, 336, 358, 
406, 480, 483, 449, 484. 

Decimal, x. 

Denominate number, xii. 

Division, 285, 288, 307, 308, 314, 
315, 333, 334, 335, 336, 351, 
352, 358, 406, 408, 4382, 433, 
439, 449, 454, 482, 485, 492. 

Fraction, 266, 271, 324, 564, 482, 
433, 484, viii. 

_ Interest, 445, 446, 487, xvili. 

Multiplication, 228, 285, 287, 
307, 314, 332, 333, 385, 336, 
341, 358, 406, 408, 433, 450, 
454, 466. 

Numbers, reduction of, 202, 
227, 228, 245, 261, 270, 
286, 311, 326, 327, 328, 
335, 336, 338, 841, 356, 
406, 408, 483, 441, 444, 
475, 492, 404. 

Subtraction, 208, 285, 287, 
Abpapre WAnbrhs Gainritey 
358, 430. 

Concentric circles, 572, 592. 

Concrete number, i. 

Cone, definitions, etc., 594. 

Surface of, 512, 518, 595, 
598. 

Developed surface, 512, 513, 
595, 596. 

Volume of, 513, 514, 600, 604, 
605. 

Frustum of, 596. 

Surface of, 597. 
Volume of, 603. 

Connecticut rule, partial payments 

(see Manual), 611, 612, 613. 


203, 
285, 
329, 
3577, 
445, 


288, 
348, 


INDEX. 


Consequent, 466, xxi. 

Consignee, Xv. 

Consignment, Xv. 

Consignor, Xv. 

Construction, of angles, 559, 
561, 562, 563, 564, 565, 
567, 569, 577, 578, 579, 
581, 582, 585. 

Of parallel lines, 562, 563, 564. 

Of parallelograms, 366, 569. 

Of rectangle, 366, 367, 570, 571, 
578, 584, 592. 

Of regular polygon, 574, 576, 
577, 578, 579, 590, 591. 

Of rhomboid, 366, 867, 570, 578. 

Of rhombus, 3866, 867, 569, 578, 
579, 588. 

Of square, 569, 573, 574, 576, 
577, 578, 579, 583. 

Of trapezium, 366, 570, 571. 

Of trapezoid, 366, 569, 570, 571. 

Of triangle, acute-angled, 365, 
571, 584. 

Equilateral, .866, 566, 567, 
574, 575, 577, 579, 580, 582, 
583, 584. 

Isosceles, 366, 867, 567, 575, 
578, 579, 582, 584. 

Obtuse-angled, 571, 
5838, 584. 

Scalene, 366, 564, 576, 578, 
579, 580, 582, 584, 585, 586. 

Right-angled, 365, 367, 566, 
567, 578, 579, 580, 581, 584, 
586. 

Convex surface, of cone, 512, 518, 
595, 598. 

Of cylinder, 511, 593, 595. 

Of hemisphere, 599. 

Of frustum, of cone, 597. 
Of pyramid, 596. 

Of prism, 511, 512, 592, 595. 

Of pyramid, 512, 595, 596. 

Of solids, development of, 511, 
512, 518, 594, 595, 596, 597. 

Couplet, xxi. 

Creditor, xi. 

Cross-addition, 211, 216, 2'75, 302, 
510. 

Subtraction, 339. 
Cube (third power), 519. 


560, 
566, 


580, 


365, 


INDEX. 625 


Cube, Continued. 
Definition, 519, xxiii. 
Root, 507, 519, 520, 522. 
Of decimals, 520, 522. 
Of fractions, 520, 
Cubie measure, 361, 402, 607. 
Customs (see Duties), xvi. 
Cylinder, definition, etc., 593. 
Surface of, 511, 512, 598, 595. 
Volume of, 513, 599, 604. 


Dates, time between, 345, 346, 347, 
348, 398, 404, 458, 459, 404, 
498, 609, xiii. 

Days of grace, 402, 4038, 451, 458, 
609. 

Debtor, xi. 

Decagon, 573. 

Decimals, addition of, 241, 283, 
292, 440, 441, 482. 

Cube root of, 520, 522. 

Definitions, etc., x, xi. 

Division of, 244, 279, 280, 281, 
282, 283, 293, 294, 295, 306, 
364, 480, 440, 441, 442, 482. 

Miscellaneous, 324, 325, 364, 440, 
441. 

Multiplication of, 242, 243, 278, 
279, 282, 283, 293, 295, 306, 
440, 

Notation and numeration of, 
238, 239, 440, 441, 483. 

Reduction of, 291, 292, 296, 440, 
el eo. 

Square root of, 481, 482, 489. 

Subtraction of, 242, 283, 293, 
364, 440, 441. 

Definitions, ete., i, xxvi. 

Abstract number, i. 

Acceptance, 451, xxvi. 

Account current, xii. 

Accurate interest, 494. 

Acute angle, 365, 557. 

Acute-angled triangle, 565. 

Addends, ii. 

Addition, ii. 

Adjacent angles, 365, 557. 

Ad valorem duties, xvi. 

Altitude, of parallelogram, 366, 
569. 

Of prism, 593. 


Definitions, ete., Continued. 


Of pyramid, 594. 
Of triangle, 569. 

Amount, addition, ii. 
Interest, 394, xvi. 
Percentage, xiii. 

Angles, 547. 

Annual interest, xviii. 

‘Antecedent, 466, xxi. 

Arabic notation, i. 

Are, 572. 

Assets, xxii. 

Avails (see Proceeds). 

Average, term of credit, xxi. 
Time, 500. 

Balance, xil. 

Bank discount, xix. 

Barrel, 607. 

Base, bonds, xxv. 
Commercial discount, xiv. 
Commission, xv. 

Duties, xvi. 

Insurance, xv. 

Line (U.S. lands), 524. 

Of isosceles triangle, 366, 567. 
Of prism, 592. 

Percentage, 383, xiii. 

Profit and loss, 386, xiv. 
Stocks, xxv. 

Taxes, XVi. 

Bill, xii. 

Of exchange, domestic, 448, 
xii. 
Foreign, 448, xii. 

Board foot, 514. 

Bonds, xxv. 

Broker, xxv. 

Brokerage, xxv. 

Cancellation, v. 

Capital, xxii. 

Stock, xxiv. 

Check, xxvi. 

Chord, 572. 

Circles, great, 518, 598. 
Small, 598. 

Circumference, 571. 

Coefficient, 530. 

Commercial discount, xiv. 

Commission, Xiv. 

Agent, xiv. 
Common denominator, ix. 


626 


Definitions, etce., Continued. 

Factor, v. 

Fraction, vii. 

Multiple, vi. 

Prime factor, v. 
Complex, decimal, x. 

Fraction, viii. 
Composite, factor, v. 

Number, v. 
Compound, decimal, x. 

Denominate number, xii. 

Fraction, viii. 

Interest, xviii. 
Concentric circles, 572. 
Concrete number, i. 
Cone, 594. 

Consequent, 466. 
Consignee, Xv. 
Consignment, xv. 
Consignor, Xv. 

Convex surface, 511, 592. 
Couplet, xxi. 

Creditor, xi. 


Cube (third power), 519, xxiii. 


toot, 519, xxiii. 

Cylinder, 593. 
Debtor, xi. 
Decagon, 573. 

Decimal, x. 

Fraction, x. 

Point, x. 

Demand note, xxv. 
Denominate, fraction, xii. 

Number, xii. 

Unit, xii. 
Denominator, 203, vii. 
Diagonal, 576. 
Diameter of circle, 571. 
Difference, subtraction, ii. 

Percentage, xiii. 
Discount, stocks, xxiv. 
Dividend, division, iv. 

Stocks, 442, xxv. 
Division, iv. 

Divisor, iv. 

Domestic exchange, xx. 
Draft, 448, xxvi. 
Drawee, xxvi. 

Drawer, xxvi. 

Duties, xvi. 

Endorser, xxv. 


INDEX. 


Definitions, etc., Continued. 


Equal triangles, 585. 
Equated time, 500, xx. 
Hquation of payments, 500, xx. 
Equilateral triangles, 366, 566. 
Equivalent triangles, 585. 
Even number, v. 
Evolution, xxiii. 
Exact, divisor, v. 
Interest, 494. 
Exchange, domestic, xx. 
Foreign, xx. 
Exponent, 547, xxiii. 
Exterior angle, 565. 
Extremes, 477. 
Face of note, xxv. 
Factor, v. 
Factoring, v. 
Fire insurance, xv. 
Foreign, bills of exchange, 448. 
Exchange, xx. 
Fraction, vii. 
Fractional unit, vii. 
Frustum, 596. 
Great circle, 518, 598. 
Greatest common, divisor, 252, v. 
Factor, v. 
Measure, v. 
Gross, 259. 
Heptagon, 573. 
Hexagon, 573. 
Horizontal line, 560. 
Improper fraction, viii. 
Indorser, xxv. 
Insurance, xv. 
Insured, xv. 
Insurer, xv. 
Interest, xvi. 
Interest-bearing note, xxv. 
Invoice, xii. 
Involution, xxiii. 
Isosceles triangle, 566. 
Joint and several note, xxvi. 
Joint note, xxv. 
Least common denominator, 
203, 255, ix. 
Multiple, 205, vii. 
Legal rate of interest, xvi. 
Liabilities, xxii. 
Life insurance, xv. 
Like numbers, i. 


INDEX. 627 


Definitions, etc., Continued. 


Lowest terms, 202 
Maker of note, xxv. 
Marine insurance, xv. 
Market value of stock, etc., xxiv. 
Maturity of note, xxvi. 
Means, 477. 
Measure, xii. 
Minuend, ii. 
Mixed, decimal, x. 

Number, 199, viii. 
Multiple, 204, vi. 
Multiplicand, iii. 
Multiplication, iii. 
Multiplier, iii. 
Negotiable note, xxvi. 
Net, price, xiv. 

Proceeds, xv. 
Nonagon, 573. 
Non-negotiable note, xxvi. 
Notation, i. 

Arabic, i. 

Roman, i. 
Number, i. 
Numeration, i. 
Numerator, 203, vii. 
Oblique angle, 557. 
Oblique-angled triangle, 565. 
Oblique, line, 560. 

Solid, 593. 
Oblong, 568. 
Obtuse angle, 365, 557. 
Obtuse-angled triangle, 565. 
Octagon, 573. 
Odd number, v. 
Opposite angles, 557. 
Orders of units, i. 
Parallel lines, 562. 
Parallelogram, 366, 568. 
Parallelopipedon, 593. 
Partial payments, xix. 
Partners, xxii. 
Partnership, xxii. 
Par value, xxiv. 
Payee, xxv, xxvi. 
Pentagon, 573. 
Pentagonal prism, 593. 
Per cent, 316, xiii. 
Percentage, xiii. 
Period, i. 
Perpendicular lines, 561. 


Definitions, ete., Continued. 


Personal property, xvi. 
Lax; xvi 
Place of figure, i. 
BONG, ORY: 
Poll tax, xvi. 
Polygon, 578. 
Power, xxiii. 
Premium, insurance, xv. 
Stocks, etc., xxv. 
Present worth, 460. 
Prime, factor, v. 
Numbers, 251, v. 
Principal, xvi. 
Principal meridian, 524. 
Prism, 592. 
Proceeds, xix. 
Product, iii. 
Profit or loss, xiv. 
Promissory note, xxv. 
Proper fraction, viii. 
Proportion, 476. 
Property, xvi. 
Tax, Xvi. 
Protractor, 559. 
Pyramid, 594. 
Quadrangular prism, 593. 
Quadrant, 572. 
Quadrilateral, 866, 568. 
Quotient, iv. 
Radius, 571. 
Range (U.S. lands), 524. 
Rate, xiii, xvi. 
Ratio, 466, xxi. 
Real property, xvi. 
Receipt, xii. 
Reciprocal, of a fraction, viii. 
Of a number, viii. 
Rectangle, 366, 568. 
Reduction, ascending, xiii. 
Descending, xii. 
Of denominate numbers, xii. 
Of fractions, viii. 
Regular, polygon, 573. 
Solid, 593. 
Remainder, ii. 
Rhomboid, 366, 568. 
Rhombus, 366, 568. 
Right angle, 365, 491, 557. 
Right-angled triangle, 565. 
Right solid, 593. 


28 INDEX. 


Definitions, etc., Continued. 
Root, xxiii. 

Roman notation, i. 
Scalene triangle, 366, 566. 
Secant line, 582. 

Second power, xxiii. 
Section (U.S. lands), 524. 
Sector, 572. 

Segment, 516, 572. 
Sextant, 572. 

Share (stock), xxiv. 
Sight draft, xxvi. 


Simple, denominate number, xii. 


Fraction, viii. 
Slant height, 594. 
Small circle, 598. 
Specific duty, xvi. 
Sphere, 598. 
Square, 568. 
(Second power), 463, xxiii. 
Root, xxiii. 
Stock, xxiv. 
Stockholder, xxiv. 
Subtraction, ii. 
Subtrahend, ii. 
Sum, ii. 
Discounted, xix. 
Tangent, circles, 572. 
Line, 582. 
Taxes, Xvi. 
Term of discount, 403, xix. 
Terms of fraction, vii. 
Third power, xxiii. 
Time, draft, xxvi. 
Note, xxv. 
Township (U.S. lands), 524. 
Transposing, 377. 
Trapezium, 366, 568. 
Trapezoid, 366, 568. 
Triangular prism, 593. 
True discount, 460. 
Underwriter, xv. 
Unit, i. 
Of fraction, vii. 
Unlike numbers, i. 
Value of fraction, vii. 
Vertical, angles, 557. 
Line, 560. 
Degrees of longitude, length of, 
450, 451, 454, 507, 598. 
Demand notes, 394, 458, 496, 


| Demand notes, Continued. 


612, 618, 614, 615, 616, 617, 
xx Ve 

Denominate fraction, 228, 285, 311, 
327, 328, 329, 433, 484, xii. 

Denominate numbers, addition of, 
236, 270, 285, 287, 312, 329, 
330, 336, 358, 406, 480, 453, 
449, 484, 

Definitions, ete., xii. 

Division of, 285, 288, 307, 308, 
314, 315, 3833, 334, 535, 536, 
351, 352, 358, 406, 408, 453, 
439, 449, 454, 482, 483, 492, 
510. 

Metric system, 525, 526, 527, 528, 
529. 

Miscellaneous, 227, 300, 301, 307, 
308, 309, 328, 329, 335, 336, 
341, 351, 352, 358, 362, 364, 
406, 408, 483, 482, 483, 484, 
492. 

Multiplication of, 228, 285, 287, 
314, 332, 338, 335, 336, 341, 
358, 406, 408, 433, 450, 454, 
466. 

Reduction of, 202, 208, 227, 228 
245, 261, 270, 285, 286, 287, 
311, 326, 327, 328, 329, 335, 
336, 338, 341, 356, 357, 406, 
408, 433, 441, 444, 445, 475, 
492, 494. 

Subtraction of, 208, 285, 287, 288, 
312, 331, 332, 335, 336, 348, 
358, 430. 

Denominate unit, 284, xii. 
Denominator, 203, 548, vii. 
Least common, 203, 249, 255, ix. 
Developed surface, of cone, 512, 
513, 595, 596. 

Of cube, 290. 

Of frustum, of cone, 596, 597. 
Of pyramid, 596, 603. 

Of prism (see Manual), 345, 356, 
402, .434, 511, 594. 

Of pyramid, 512, 595, 596, 600, 
603. 

Diagonal, of polygon, 576, 579. 

Of rectangle, 322, 491, 504. 

Of rhomboid, 581. 

Of rhombus, 496, 504. 


INDEX. 


Diagonal, Continued. 

Of square, 322, 491, 503, 504. 

Of trapezium, 495, 504, 508. 

Diameter, of circle, 450, 468, 469, 
503, 571, 592. 

Of sphere, 518, 519, 597, 599, 
605. 

Difference, between dates, 345, 346, 
347, 348, 398, 404, 458, 459, 
494, 498, 609, xiii. 

Subtraction, ii. 

Percentage, xiii, xiv, Xv, XXv. 

Discount, bank, 402, 403, 404, 407, 
408, 420, 421, 423, 424, 425, 
455, 470, 4838, 494, xix. 

Commercial, 410, 411, 421, 422, 
431, 446, 456, 457, 475, 479, 
480, 481, xiv. 

Exchange, 448, 449, 451, 452, 
485, Xx. 

Stocks and bonds, xxv. 

True, 460, 494. 

Distances, calculation of, 587, 588, 
589, 590. 

Dividend, division, 235, 236, iv. 

Stocks, 442, 448, xxv. 

Divisibility of numbers, 252. 

Division, of decimals, 244, 279, 280, 
281, 282, 288, 298, 294, 295, 
306, 364, 480, 440, 441, 442, 
482. 

Rule, 280, xi. 

Of denominate numbers, 285, 
288, 307, 308, 514, 3815, 333, 
334, 335, 336, 851, 352, 358, 
406, 408, 483, 489, 449, 454, 
482, 483, 492, 510. 

Of fractions, 209, 214, 215, 216, 
250, 256, 267, 268, 269, 271, 
278, 296, 307, 310, 324, 363, 
364, 427, 481, 482, 483, 483, 
484, 

Rules, 267, x. 
Of integers, 235, 2386, 263, 341, 
400, 480, 465. 
Principles and rule, iv. 

Of lines into equal parts (see 
Bisection ), 581, 587. 

Divisor, 236, iv. 

Domestic exchange, 448, 449, 451, 
452, 485. 


629 


Domestic exchange, Continued. 
Definitions, etc., 447, 448, xx, 
=x Vi 
Drafts, sight, 448, 449, 452, 485. 
Time, 451, 452, 485. 
Acceptance of, 451. 
Definitions, xx, xxvi. 
Drawee, xxvi. 
Drawer, xxvi. 
Drills, sight, 
307, 337, 
505. 
Dry measure, 227, 607. 
Duties, 290, 300, 317, 352, 384, 466, 
515. 
Definitions, etc., xvi. 


229, 232, 308, 306, 
881, 396, 426, 474, 


Endorsement (see Manual), 458, 
459, 460, 496, 611, 612, 618, 
614, 615, 616, 617, xix. 

Endorser, xxv. 

Liability of, xxv. 


English money, 408, 480, 482, 441, 


442, 455, 456, 466, 482, 483, 
484, 492, 510. 
Tables, 408, 608. 
Equal triangles, 522, 584. 
Equated time, 500, xx. 
Equation of payments, 499, 500, 
501. 


Definitions, etc., 500, xx. 
Equations (see Algebraic equa- 
tions), 369, 530. 
Equilateral triangle, altitude of, 
496, 585, 591. 
Area of, 496, 503, 504, 600. 
Construction of, 570, 574, 575, 
577, 579, 580, 581, 582, 583, 
584. 
Definition, 566. 
Equivalent triangle, 585. 
Erection of perpendicular, at any 
point in line, 560, 577. 
At extremity of line, 577. 
At middle of line, 575. 
Through point outside of line, 
579. 
Even numbers, v. 
Evolution, 468, 519. 
Definitions, etc., 468, 519, xxiii. 
Exact divisor, v. 


630 


Exact interest, 494, 495. 
Rule, 3tc., xviil. 
Exchange, domestic, 448, 449, 451, 
452, 485. 
Foreign, 455, 456, 457. 
Definitions, 447, 448, xx. 
Exponent, 463, 519, xxiii. 


Face of bill of exchange, to find, 
456. 
Of draft, to find, 448, 449, 452, 
485, xx. 
Of note, xxv. 
To find, 424, 425, 470, 494, 
ab ah 
Face value of stocks and bonds, 
388, 443, 488, xxiv. 
Factors, 204, 205, 250, 251, 291, 
507, v. 
Prime, 291, 507, v. 
Federal money, 212, 216, 217, 218, 
224, 429, 
Fractional parts of a dollar, oral, 
218, 219, 220, 221, 222, 223, 


924, 295, 272, 278, 299, 300, 


303, 304, 335, 397, 3898, 427, 
447, 455, 475, 507. 

Fence, area of, 289, 348, 354, 390, 
435, 516. 

Fire insurance, 317, 364, 384, 385, 
386, 423, 444. 

Definitions, etc., xv. 

Foreign coins, value of, 619. 

Foreign exchange, 448, 455, 456, 
457, Xx. 

Fractional unit, vii. 

Fractions, addition of, 199, 200, 
201, 202, 208, 204, 205, 249, 
255, 262, 277, 309, 362, 364, 
432, 453, 483, 484. 

Clearing of, 373, 477. 

Complex, 266, 268, 271, 324, 364, 
433, 484. 

Compound, 266, 271, 824, 364, 
432, 433, 484. 

Cube root of, 520, xxiv. 

Decimal, x. 

Definitions, principles,and rules, 
Vile vail Seen ke 

Denominate, 228, 285, 311, 327, 
328, 329, 4383, 484. 


INBEX., 


Fractions, Continued. 


Division of, 209, 214, 215, 216, 
250, 256, 267, 268, 269, 271, 
278, 296, 307, 310, 324, 363, 
364, -427, 431, 482, 433, 483, 
484, 

Miscellaneous, 205, 207, 208, 209, 
250, 256, 266, 268, 310, 324, 
364, 482, 483, 483, 484. 

Multiplication of, 209, 213, 214, 
218, 250, 256, 265, 266, 268, 
277, 282, 307, 310, 323, 340, 
368, 364, 399, 405, 482, 433, 
A471, 484. 

Reduction of, 199, 200, 201, 202, 
203, 205, 252, 253, 256, 257, 
9258, 284, 325, 482, 484. 

Square root of, 478, xxiv. 

Subtraction of, 205, 206, 207, 
250, 256, 262, 268, 277, 296, 
310, 324, 363, 364, 482, 433, 
483, 484. ; 

French money, 456, 476, 478, 608. 
Frustum of cone, 596. 
Developed surface, 597. 
Surface, 597. 
Volume, 603. 

Of pyramid, 596, 597. 
Developed surface, 596, 6038. 
Surface, 596, 597. 

Volume, 601, 602. 
Furlong, long measure, 607. 


Geometrical exercises, 365-367, 
557-605. 

Angles, 365, 557, 558, 559, 560, 
561, 562, 568, 564, 565, 566, 
567, 569, 577, 578, 579, 580, 
581, 588, 585, 590. 

Calculating heights and dis- 
tances, 587, 588, 589, 590. 

Circles, 571, 572, 578, 574, 575, 
576, 577, 579, 580, 582, 583, 
584, 590, 591, 592, 598. 

Construction problems, 575-584, 

Mensuration (see Measure- 
ments ), 600-605. 

Parallels, 562, 563, 564, 565, 587, 
588, 589, 590. 

Polygons, 5738, 574, 576, 577, 579, 
584, 591, 592. 


INDEX. 


Geometrical exercises, Continued. 
Quadrilaterals, 3866, 367, 568, 
569, 570, 571, 574, 577, 578, 
579, 580, 581, 582, 583 584, 
590. 

Triangles, 366, 367, 564, 565, 
566, 567, 570, 5738, 574, 575, 
576, 577, 578, 579, 580, 581, 
582, 583, 584, 586, 587, 588, 
589, 590. 

German money, 442, 456, 476, 608. 
Government lands, 524, 525. 
Grace, days of, 402, 403, 451, 485, 
609. 
Great circle, 518, 522, 598. 
Greatest common divisor, 252, 253, 
258, 482, v, vi. 
Factor, v. 
Measure, v. 


Heights, calculation of, 436, 587, 
588, 589, 590. 
Heptagon, 573, 574. 
Hexagon, 504, 578. 
Apothem of regular, 591. 
Area of, 504, 591. 
Construction of, 578, 574, 579, 
584. 
Perimeter of, 591. 
Horizontal line, 560. 
Hypotenuse, 486, 490, 491, 496, 
506, 525, 526, 555, 556, 580, 
581, 596. 


Improper fraction, 266, 267, viii, 
ja. 
Indorsement, xix. 
Indorser, Xxv. 
Inscribed polygon, 578, 574, 576, 
577, 579, 591, 594. 
Square, 504, 516, 574, 576, 577. 
Triangle, equilateral, 574, 577, 
579, 582. 

Right-angled, 579, 580, 582. 
Scalene, 579, 582, 585. 
Insurance, 317, 364, 384, 385, 386, 

423, 444. 
Definitions, etc., xv. 
Interest, 320, 321, 349, 350, 351, 
352, 353, 384, 3886, 393, 394, 
395, 398, 404, 407, 408, 415, 


631 


Interest, Continued. 
416, 417, 418, 419, 420, 421, 
423, 431, 488, 442, 443, 445, 
446, 458, 459, 460, 469, 470, 
482, 487, 493, 494, 495, 496, 
497, 498, 528, 610, 613, 614, 
615. 

Accurate, 494, 495. 

Annual, 528, 613, 614, 615, xviii. 

Compound, 445, 446, 487, xviii. 

Definitions, etc., xvi, xvii, xviii, 
xi 

Exact, 494, 495, xviii. 

General method of computing, 
301, 393, Xvii. 

Legal rates of, 610. 

Method by aliquot parts, 417, 
418, 419, 420, 469, 470, xvii. 

Notes bearing interest, 394, 395, 
407, 408, 458, 459, 460, 523, 
So aia 

Partial payments, 458, 459, 
460, 611, 612, 618, 614, 615, 
GIG, Olidy xix. 

Six per cent method, xviii. 

To find principal, rate, or time, 
384, 3886, 395, 415, 416, 417, 
470, 482, 489, 4938, 494, xvii. 

Invoice, 466. 

Definitions, xii. 

Involution, 463, 464, 519. 

Definitions, etc., 463, 519, xxiii. 

Isosceles triangle, altitude of, 495, 
567, 575, 578, 584, 594. 

Area of, 367, 411, 412, 495, 496, 
503. 

Base of, 567. 

Construction of, 366, 367, 567, 
575, 576, 578, 579, 582, 584. 

Definitions, 366, 566. 


Joint and several note, 611, xxvi. 
Joint note, xxv. 


Leap years, 608. 
Least common denominator, 203, 
249, 255. 
Definitions, etc., 205, 255, ix. 
Least common multiple, 205, 249, 
254, 255, 432, 433. 
Definitions, etc., 205, vi, vii. 


632 


Ledger account (see Manual), 
ya 

Legal rates of interest, 610. 

Liabilities, xxii. 

Liability of indorser, xxv. 

Life insurance, Xv. 

Like numbers, i. 

Lines, bisection of, 575. 

Division of, into equal parts, 
581, 587. 

Horizontal, 560, 561, 562, 563, 
576, 581. 

Oblique, 560, 561, 562, 563, 567, 
575, 576, 577, 578, 581. 

Parallel, 562, 563, 564. 

Perpendicular, 365, 561, 577,579. 

Tangent, 582. 

Secant, 582. 

Vertical, 660. 

Link, 441, 486, 607. 

Liquid measure, table, 227, 607. 

Long division (see Division). 

Drills, 232. 

Longitude and time, 452, 453, 454, 
471, 472, 478. 

Length of degrees of longitude, 
450, 451, 454, 507, 598. 

Long measure, 285, 607. 

Long ton, 541, 608. 

Lowest terms, 202, 203; 205, 252, 
253, 256, 257, 258,, 284, 292, 
296, 325, 432, 484, viii. 

Lumber measure, 514, 515. 


Market value of stocks and bonds, 
SOKA Vs 

Maturity of note (see Days of 

. grace), 403, xxvi. 

Means, 477. 

Measure, xii. 

Measurements, miscellaneous, 342, 
343, 344, 345, 358, 354, 355, 
306, 358, 359, 360, 389, 390, 
400, 401, 402, 411, 412, 413, 
414, 488, 484, 435, 486, 461, 
462, 490, 491, 492, 493, 495, 
496, 508, 504, 511, 512, 513, 
514. 

Of arcs, 450, 451, 572, 577, 580, 
590, 598. 
Of angles, 558, 559, 560, 561, 


INDEX. 


Measurements, Continued. 
562, 563, 564, 565, 566, 567, 
569, 577, 579, 580, 

Of bins, 359, 360, 361, 435, 461. 

Of boards, 434, 514, 515, 516. 

Of carpeting, 301, 345, 389, 401. 
434, 461, 462, 526. 

Of chords, 516, 572, 582, 590, 
598. 

Of circles, 461, 496, 5038, 504, 
516, 517, 526, 592. 

Of circular rings, 517, 592. 

Of circumference, 450, 468, 469, 
503. 

Of cone, 512, 513, 514, 595, 596, 
598, 600, 604, 605. 

Of cylinder, 511, 512, 514, 519, 
522, 600. 

Of degrees of longitude, 450, 451, 
507, 598. 

Of fencing, 289, 343, 354, 389, 
400, 401, 4385. 

Of frustum, 596, 597, 601, 602, 
603. 

Of hexagon, regular, 504, 591. 

Of hypotenuse, 486, 491, 496, 
506, 525, 526, 555, 556, 580, 
581, 596. 

Of lumber, 514, 515. 

Of octagon, regular, 591. 

Of parallelopipedon, 290, 316, 
345, 358, 359, 360,. 362, 390, 
414, 434, 461, 462, 493, 527. 

Of prism, 511, 512, 513, 595, 599, 
600. ; 

Of pyramid, 512, 513, 595, 596, 
600, 601, 602, 604. 

Of rectangle, 246, 247, 248, 262, 
276, 288, 284, 286, 288, 289, 
290, 297, 300, 301, 315, 342, 
348, 344, 345, 353, 354, 355, 
356, 366, 367, 3889, 390, 401, 
411, 412, 413, 427, 431, 485, 
436, 461, 462, 475, 476, 482, 
485, 491, 504, 506, 526, 550, 
555, 584. 

Of regular polygon, 591. 

Of rhomboid, 367, 412, 418, 503, 
584. 

Of rhombus, 3867, 412, 413, 496, 
504, 


INDEX. 


Measurements, Continued. 

Of rooms, 289, 315, 316, 345, 359, 
389, 390, 400, 434, 461, 462. 

Of sector, 516, 592. 

Of segment, 516. 

Of sphere, 518, 519, 521, 522, 
599, 605. 

Of tanks, 360, 389, 402, 406, 433, 
435, 461, 482, 514, 527. 

Of trapezium, 367, 368, 413, 495, 
504, 508, 556. 

Of trapezoid, 355, 367, 368, 412, 
413, 484, 490, 491, 504. 

Of triangle, right-angled, 322, 
328, 355, 367, 411, 412, 433, 
461, 486, 490, 491, 496, 526, 
551, 555, 584, 585, 586. 

Oblique-angled, 356, 367, 
411, 412, 461, 495, 496, 503, 
504, 508, 556, 584, 585, 586. 
Of wood, 359, 402, 485, 527. 
Measures, circular, 608. 

Of capacity, 227, 361, 402, 607. 

Of extension, 285, 607. 

Of time, 226, 608. 

Of value, 408, 608. 

Of weight, 227, 361, 406, 607, 
608. 

Mensuration (see Measurements ). 

Merchants’ rule, partial payments, 
496, 497, 498. 

Metric system, 525, 526, 527, 528, 
529. 

Minuend, 235, ii. 

Mixed decimal, x. 

Mixed numbers, 199, 257, 266, 267, 
viii, ix. 

Multiple, 204, 205, 249, vi, vii. 

Least common, 205, 254, 255, 
432, 433, vi, vii. 

Multiplicand, 235, 300, iii. 
Multiplication of algebraic quan- 
tities, 547, 548, 549. 

Of decimals, 242, 243, 278, 279, 
282, 283, 293, 295, 306, 364, 
440, 441. 

Rule, xi. 

Of denominate numbers, 228, 
285, 287, 314, 332, 338, 335 
336, 341, 358, 406, 408, 433, 
450, 454, 466. 


6338 


Multiplication, Continued. 

Of fractions, 205, 209, 218, 214, 
218, 250, 256, 265, 266, 268, 
277, 282, 307, 310, 328, 340, 
363, 364, 399, 405, 432, 433, 
471, 484. 

Principles and rule, viii, ix. 

Of integers, 234, 303, 305, 307, 
340, 428, 429, 471. 

Principles, definition, and 
rule, iii, iv. 
Cross-multiplication, 212. 

Short methods, 229, 260, 276, 
298, 303, 305, 323, 337, 340, 
363, 396, 399, 426, 428, 429, 
471. 

Multiplier, 235, 300, iii. 


Negotiable note, 394, 402, 407, 458, 
460, 496, 523, 611, 612, 618, 
614. 

Definition, xxvi. 

Net. price, xiv. 

Net proceeds, xv. 

New Hampshire rule, partial pay- 
ments, 614, 615. 

Nonagon, 573. 

Construction of regular, 574, 
590. 

Non-negotiable note, 615, 616, 617, 
XXvi. 

Notation of decimals, 239, 240, 
441, x. 

Of integers, 210, i. 
Roman, i. 
Notes (see Manual), xxv, xxvi. 
Demand, 394, 458, 496, 612, 613, 
614, 615, 616, 617, xxv. 
Interest-bearing, 394, 407, 458, 
460, 496, 611, 612, xxv. 
Annual interest, 523, 613, 
614, 615, 616, 617, xviii. 
Joint, xxv. 
Joint and several, 611, xxvi. 
Negotiable, 394, 402, 407, 458, 
460, 496, 523, 611, 612, 613, 
614, xxvi. 
Non-negotiable, 615, 616, 617, 
XXVi. 
Time, 402, 407, 460, 528, 611, 
612, xxv, 


634 


Notes, Continued. 
Days of grace, 402, 403, 451, 
485, 609. 
Numeration of decimals, 239, 240, 
440, 441, 483, x. 
Of integers, 210, 211, i, ii. 


Oblique angle, 557, 558. 

Cone, 594, 604. 

Cylinder, 593, 604. 

Line, 560, 561, 562, 563, 567, 
575, 576, 577, 578, 581. 

Prism, 593, 604. 

Pyramid, 594, 604. 

Oblique-angled triangle, 565. 

Area of, 356, 367, 411, 412, 461, 

495, 496, 503, 504, 508, 556. 
Oblong (see R ectangle), 568. 
Obtuse angle, 365, 557, 559, 561, 

565, 566. 

Obtuse-angled triangle, 365, 565. 

Altitude of, 569, 578, 584. 

Area of, 412, 570, 571, 584, 585. 

Construction of, 571, 583, 584. 

Octagon, 573. 

Apothem of regular, 591. 

Area of, 591. 

Construction of, 574, 576, 579. 

Perimeter of, 591. 

Odd numbers, v. 
Opposite angles, 557, 561. 
Oral, exercises, 199, 200, 201, 205, 

206, 214, 218, 219, 22 7 221) 

‘ 222, 223° 224, 239, 250, 
267, 278, 284, 316, 317, 
328, 349, 369, 378, 382, 385, 
386, 417, 468, 466, 467, 
480, 489. 

Problems, 202, 203, 230, 
236, 257, 258, 261, 269, 
272, 2738, 276, 284, 299, 
308, 304, 807, 308, 818, 335, 
338, 345, 346, 351, 352, 
397, 398, 411, 423, 427, 
436, 446, 447, 452, 453, 454, 
455, 467, 468, 475, 476, 
478, 491, 492, 499, 406, 507. 

Orders of units, 308, i. 


Par sixk. 
Parallel lines, 562. 


INDEX. 


Parallel lines, Continued. 
Construction of, 562, 5638, 564. 
Parallelogram, altitude of, 366, 

569. 

Angles of, 569, 590. 

Area of (see Rectangle, Rhom- 
boid, etc. ), 412, 604. 

Construction of, 366, 569, 570, 
574, 577, 578, 579, 580, 582, 
5838, 584. 

Definitions, 366, 568. 

Parallelopipedon, definition, 593. 

Surface of, 290, 316, 345, 354, 
356, 389, 406, 418, 414, 434. 

Volume of, 358, 359, 360, 362, 
390, 402, 418, 414, 435, 454, 
461, 462, 493. 


Partial payments, Connacnent 
rule (see Manual), 611, 612, 
6138. 

Merchants’ rule, 496, 497, 498, 
ah 

United States rule, 458, 459, 460, 
2.8 Da 


Annual interest notes, 613, 614. 

New Hampshire rule, 614, 
615. 

Vermont rule, 615, 616, 617. 

Partnership, 352, 407, 409, 439, 
440, 454, 455, 482, 483, 501, 
502, 508, 509. 

Definitions, ete., xxii. 

Par value, 388, 442, 448, 444, 488, 
Xxiv. 

Payments, partial (see Partial pay- 
ments), Xix, 

Equation of (see Equation of 
payments), xx. 

Pentagon, 573. 

Construction of regular, 
574. 

Percentage, 316, 317, 349, 351, 352, 
361, 881, 382, 383, 384, 385, 
386, 387, 388, 3898, 407, 408, 
410, 411, 421, 422, 423, 
431, 482, 442, 444, 446, 
449, 456, 466, 479, 480, 
482, 483, 488, 489, 491, 492, 
494, 506. 

Bonds, 442, 
489, Xxv. 


443, 444, 445, 488, 


INDEX. 


Percentage, Continued. 

Commercial discount, 410, 411, 
421, 422, 431, 446, 456, 457, 
466, 475, 479, 480, 481, xiv. 

Commission, 317, 384, 385, 398, 
423, 449, 482, xiv. 

Definitions, etc., 316, 381, xiii, 
xiv. 

Duties, 317, 352, 384, 466, xvi. 

Insurance, 317, 564, 385, 386, 
423, 444, xv. 

Profit and loss, 817, 818, 386, 
387, 388, 391, 392, 423, 432, 
446, 447, 483, 491, 492, 494, 
xiv. 

Stocks, 388, 442, 448, 444, 446, 
488, 489, xxiv, xxv. 

Taxes, 317, 385, 444, 446, xvi. 

To find the base or the rate, 383, 
384, 385, 386, xiii, xiv. 

Perimeter of rectangle, 299, 301, 

309, 427, 435. 

Of square, 435, 485. 

Of right-angled triangle, 491. 

Of regular polygon, 591. 

Of trapezoid, 491, 

Perpendicular lines, 865, 561, 562, 

577, 579. 

Policy of insurance, xv. 

Poll tax, 617, 618, xvi. 

Polygon (see Areas, etc. ), 5738. 
Angles of regular, 573, 574. 
Construction of regular, 573, 

574. 

Powers, 463, 519, xxiii. 

Premium, exchange, 448, 449, 451, 
452, 485, xx. 

Insurance, 385, 484, xv. 

Stocks and bonds, xxv. 

Present worth, 460. 

Prime factors, 251, 291, 507, v. 
Numbers, 251, 257, 291, v. 

Principal (see Interest), 350, 393, 

415, xvi. 

Principles, addition, ii. 
Cancellation, v. 

Decimals, x. 

Division, iv. 

Fractions, viii. 

Addition of, 203, ix. 
Subtraction of, 206, ix. 


635 


Principles, Continued. 

Greatest common divisor, vi. 

Least common multiple, vii. 

Multiplication, iii. 

Proportion, xxii. 

Ratio, xxi. 

Subtraction, ii. 

Prisms, kinds, ete., 592, 593. 

Surface of, 511, 512, 595. 

Developed surface of, 356, 
434, 511, 594. 

Volume of, 518, 514, 599, 600. 

Problems in interest, 384, 386, 395, 
415, 416, 417, 470, 482, 489, 
493, 494, xvii. 

In percentage (see Percentage, 
etc. ), 383, 384, 385, 386, 398, 
407, 430, 481, 482, 447, 492, 
XI SL Ve 

Proceeds (see Bank discount), 
402, xix. 

Product, 235, 300, iii. 

Profit and loss, 317, 318, 386, 387, 
388, 391, 392, 428, 482, 446, 
447, 483, 491, 492, 494, xiv. 

Proper fraction, viii. 

Proportion (see Analysis), 476, 
477, 478, 479. 

Definitions, etc., xxi, xxii. 

Protractor, 558. 

Pyramid, definition, etc., 594. 

Surface of, 512, 595, 596. 

Developed surface of, 512, 
595, 596, 600, 603. 

Surface of frustum of, 596, 597. 
Developed surface, 596, 603. 

Volume of, 5138, 600, 601, 604. 

Volume of frustum of, 601, 602, 
604. 


Quadrilateral (see Rectangle, 
Rhomboid, ete. ), 366, 4138, 508, 
568. 

Construction of (see Square, 
etc. ), 366, 569, 570, 571, 574, 
577, 578, 579, 580, 581, 582, 
583, 584, 590. 

Quotient, 235, 236, iv. 


Radius, of circle, 469, 496, 503, 571, 


636 


Radius, Continued. 
Of sphere, 518, 521, 598, 599, 
605. 
Range, U. S. lands, 524. 
Rate (see Discount, Interest, Per- 
centage, etc. ), xiii, xvi, xvii. 
Ratio, 466, 467, 468, 469, 516, 517, 
519, 521, 522, 582, 584, 595. 
Definitions, etc., 466, xxi. 
Rectangle, area of, 246, 247, 248, 
262, 276, 283, 284, 286, 288, 
289, 290, 297, 300, 301, 315, 
338, 342, 348, 344, 345, 352, 
358, 354, 355, 356, 389, 390, 
397, 3898, 400, 401, 411, 412, 
413, 427, 481, 485, 436, 461, 
462, 475, 476, 482, 504, 506, 
517, 525, 526, 550, 555, 584. 
Definition, 568. 
Diagonal of, 322, 491, 504. 
Perimeter of, 299, 801, 309, 427, 
435. 
Reduction of decimals to common 
fractions, 292, 296, 441, xi. 
Of denominate numbers, ascend- 
ing, 228, 231, 285, 286, 287, 
311, 326, 327, 328, 341, 356, 
357, 408, 441, 444, 445, 484, 
492, 494, xiii. 
Descending, 227, 228, 245, 
285, 286, 287, 311, 326, 327, 
328, 341, 488, 441, 484, xii. 
Of fractions to common denom- 
inator, 205, 205, 249, 255, 267, 
1x: 
To decimals, 291, 296, 351, 
364, 440, 441, xi. 
To higher terms, 200, 201, 
205, 258, viii. 
To lowest terms, 202, 203, 
205, 252, 258, 256, 257, 258, 
284, 325, 432, 484. 
To per cents, 381. 
To simplest form, complex, 
266, 268, 271, 824, 364, 433, 
483, 484, x. 
Compound, 266, 271, 324, 
' 364, 482, 433, 484, 
To whole or mixed num- 
bers, 199, 257, 266, 267, ix. 
Of mixed numbers to improper 


INDEX. 


Reduction, Continued. 
fractions, 199, 257, 266, 267, 
473, ix. 
Of per cents to fractions, 324, 
Regular polygons, area of (see 
Hexagon), 591. 

Construction of (see Pentagon, 
etc.), 575, 574. 

Definitions, 573. 

Perimeter of, 591. 

tegular, prism, 593. 
Pyramid, 594. 
Remainder, 235, iv. 
teview, bonds and stocks, 488, 
489. 

Cube root, 522. 

Decimals, 324, 825, 440, 441. 

Denominate numbers, 341, 406, 
408, 483, 441. 

Discount, bank, 420, 421, 470. 
Commercial, 479, 480, 481. 

Domestic exchange, 485. 

Fractions, 250,. 256, 277, 309, 
310, 328, 324, 340, 362, 363, 
364, 899, 405, 429, 432, 433, 
471, 483, 484. 

Fundamental processes, 211, 
212, 216, 217, 233, 234, 235, 
263, 275, 276, 302, 303, 305, 
339, 340, 341, 400, 428, 429, 
430, 465, 471, 510. 

Interest, simple, 417, 418, 419, 
420, 469, 470. 

Compound, 487. 

Longitude and time, 471, 472, 
473. 

Measurements, 288, 289, 290, 
315, 316, 342, 348, 344, 345, 
353, 354, 355, 356, 358, 359, 
360, 367, 368, 389, 390, 400, 
401, 402, 411, 412, 413, 414, 
433, 484, 435, 486, 490, 491, 
495, 496, 505, 504, 508, 516, 
517. 

Miscellaneous, 207, 208, 209, 
225, 226, 2385, 236, 287, 238, 
245, 258, 259, 270, 271, 288, 
300, 301, 3808, 309, 336, 341, 
352, 362, 364, 406, 407, 409, 
431, 482, 449, 450, 454, 475, 


INDEX. 637 


Review, Continued. 

476, 477, 478, 479, 482, 483, 
490, 491, 492, 493, 494, 501, 
502, 508, 509, 510. 

Square root, 478, 481, 489. 

Surfaces and volumes, 390, 413, 
414, 434, 435, 461, 462. 

thomboid, area of, 367, 412, 413, 
503, 570. 

Construction of, 866, 569, 570. 

Definition, 366, 567. 

Diagonals of, 581. 

Rhombus, area of, 367, 412, 413, 
496, 504, 570. 

Construction of, 366, 496. 

Definition, 366, 569, 570, 579. 

Diagonals of, 496, 504, 579. 

Right angle (see Perpendicular), 
365, 491, 557. 

Right-angled triangle, area of, 
322, 323, 355, 356, 367, 411, 
412, 433, 551, 555. 

Construction of, 566, 580. 

Definition of, 565. 

Length of sides of, 486, 490, 491, 
506, 516, 525, 555, 556, 580, 
581, 596. 

Right cone, 594. 

Cylinder, 593. 

Prism, 593. 

Pyramid, 594. 

Roman notation, i. 

Root, cube, 520, 522. 

Square, 270, 463, 464, 465, 478, 
481, 485, 486, 489, 490, 491. 

Definitions, etc., xxiii, xxiv. 

Rules, accurate interest, 494, xviii. 

Addition of algebraic quanti- 
ties, 531. 

Of decimals, xi. 
Of fractions, 203, 255, ix. 
Of integers, ii. 

Aliquot-part method of finding 
interest, 417, xvii. 

Analysis, 456. 

Angle, measurement of, 559. 

Annual interest, 523, xviii. 

Partial payments of annual- 
interest notes, 613. 
N. H. rule, 614. 
Vermont rule, 615. 


Rules, Continued. 


Area of circle, 496, 516, 592. 
Of sector, 516. 
Of trapezium, 495. 
Of triangle, 495. 
Average term of credit, xxi. 
Bank discount, 402, 405, xix. 
Of interest-bearing notes, 407. 
Proceeds, 402. 
To find face of note, ete., 424. 
Base, of right-angled triangle, 
486, 
Per cent and rate given, 383, 
Xi: 
Rate and amount given, 3853, 
xiii. 
Rate and difference given, 385, 
xiv. 
Bill of exchange, cost of, 455. 
Face of, 456. 
Board feet, 514. 
Brokerage, 488. 
Cancellation, 2638, 264. 
Cause and effect, 436. 
Circle, area of, 496, 516, 592. 
Circumference of, 450, 503, 
518. 
Diameter of, 450, 508, 518. 
Radius of, 518. 
Clearing of fractions, 374. 
Commercial discount, 410, xiv. 
Common prime factor, Vi. 
Complex fractions, reduction of, 
ib a 
Compound numbers (see De- 
nominate numbers ), xii. 
Interest, 445, 487, xviii. 
Cone, convex surface of, 595. 
Frustum of surface of, 597. 
Volume of, 603. 
Volume of, 515, 600. 
Connecticut rule, partial pay- 
ments, 611. 
Convex surface of cone, 595. 
Of cylinder, 595. 
Of frustum of cone, 597. 
Of pyramid, 597. 
Of prism, 511. 
Of pyramid, 595. 
Cube root of fractions, xxiv. 
Of integers, 520, 522, xxiv. 


638 INDEX 


Rules, Continued. 

Cylinder, surface of, 595. 
Volume of, 514, 604. 

Dates, time between, 346, 348, 
609, xiii. 

Decimals, addition of, xi. 
Division of, 280, xi. 
Multiplication of, xi. 
Reduction to common denom- 

inator, Xi. 
To common fractions, Xi. 
Subtraction of, xi. 
To read, 239, 240, x. 
To write, x. 

Denominate numbers, reduction 

ascending, 326, 357, xiii. 
Reduction descending, 326, 
xii. 

Denominator, least common, of 

fractions, 255, ix. 
Of decimals, xi. 

Diameter of circle, 450, 503, 518. 

Difference between dates, 346, 
548, 609, xiii. 

Discount, bank, 402, 403, xix. 

Of interest-bearing notes, 
407. 

Proceeds, 402. 

To find face of note, etc., 
424, 

Commercial, 410, xiv. 

True, 460. 

Dividends, stock, 488. 

Division of decimals, 280, xi. 
Of denominate numbers, 334. 
Of fractions, 214, 267, x. 

Of integers, iv. 
Draft, sight, cost of, 448, 449, xx. 
Face of, xx. 
Time, cost of, 451, xx. 
Face of, xx. 

Equated time, xxi. 

Exact interest, 494, xviii. 

Face of bill of exchange, 456. 
Of draft, xx. 

Of note, 424. 

Factors, common prime, Vi. 
Prime, v. 

Fractions, addition of, 203, 255, 

ix, 
Clearing of, 374. 


Rules, Continued. 


Complex, reduction of, ix. 
Cube root of, xxiv. 
Division of, 214, 267, x. 
Improper, reduction of, ix. 
Mixed numbers, reduction to 
improper, ix. 
Multiplication of, 265, ix. 
Reduction of, to higher terms, 
Viii. 
To least common denomi- 
nator, 255, ix. 

To lowest terms, 252, viii. 
Square root of, xxiv. 
Subtraction of, 206, ix. 

Frustum of cone, surface of, 

597. 

Volume of, 608. 

Of pyramid, surface of, 597. 

Volume of, 602. 

Greatest common divisor, 253, 

vi. 

Hypotenuse, length of, 486. 

Improper fractions, reduction of, 

ix, 

Interest, accurate, 494, xviii. 
Annual, 523, xviii. 
Compound, 445, 487, xviii. 
Exact, 494, xviii. 

Simple, general method, 350, 
393, Xvii. 

Method by aliquot parts, 
417, xvii. 

Six per cent method, xviii. 

Amount, 894. ; 

To find the principal, 415, 
XVii. 

Rate, 415, xvii. 

Time, 415, xvii. 

Least common denominator, 

255, 1x. 

Multiple, 255, vii. 

Longitude and time, 471. 

Lowest terms, reduction of frac- 
tions to, 252, viii. 

Merchants’ rule, partial pay- 
ments, 497, xix. 

Mixed numbers, reduction of, ix. 
Square root of, 4738. 

Multiple, least common, 255, 

vii. 


INDEX. 639 


Rules, Continued. 


Multiplication, algebraic quan- 
tities, 549. 
Decimals, xi. 
Fractions, 265, ix. 
Short methods, 399. 
Integers, iii. 
Short methods, 276, 308, 305, 
340. 
By 10, 100, ete., iv. 
Mixed numbers, short meth- 
ods, 276, 340, 399, 405. 
New Hampshire rule, partial 
payments, 614. 
Notation of decimals, x. 
Of integers, i. 
Numeration of decimals, x. 
Of integers, ii. 
Oblique solids, volume of, 604. 
Partial payments, Connecticut 
rule, 611. 
Merchants’ rule, 497, xix. 
United States rule, 458, 459, 
Ks 
Annual interest notes, 6138. 
New Hampshire rule, 614. 
Vermont rule, 615. 
Partnership (see Manual), xxii. 
Percentage, 383, xiii. 
To find base, 384, xiii, xiv. 
Rate, 384, xiii. 
Perpendicular right-angled tri- 
angle, 486. 
Power, second, 463, xxiii. 
Third, 519, xxiii. 
Prime factors, v. 
Common, Vi. 
Principal, to find, 415, xvii. 
Prism, convex surface, 511. 
Volume, 604. 
Proceeds, 402. 
Profit and loss, 387. 
Proportion, 477, xxiii. 
Pyramid, convex surface of 
regular, 595. 
Frustum of, convex surface, 
597. 
Volume, 602. 
Volume of, 604. 
Radius of circle, 518. 
Of sphere, 518, 599. 


Rules, Continued. 


Rate, discount, 424. 
Interest, 414, xvii. 
Percentage, 383, xiii. 
Ratio, 466. 
Reduction of decimals to com- 
mon denominator, xi. 
To common fractions, xi. 
Of denominate numbers, as- 
cending, 326, 357, xiii. 
Descending, 326, xii. 
Of fractions to the least com- 
mon denominator, 255, ix. 
To decimals, 291, xi. 
To higher terms, viii. 
To lowest terms, 252, viii. 
To simplest form, ix. 
Of improper fraction to mixed 
number, ix. 
Of mixed number to improper 
fraction, ix. 

Root, cube, 520, 522, xxiv. 
Square, 465, 481, xxiii, xxiv. 

Sphere, surface, 518, 521, 605. 
Volume, 521. 

Square root of fractions, xxiv. 
Of integers, 465, 481, xxiii. 
Of mixed numbers, 473. 

Subtraction of algebraic quanti- 

ties, 533. 
Of decimals, xi. 
Of fractions, ix. 
Of integers, ii, iii. 

Surface of cone, 595. 

Of cylinder, 595. 

Of frustum of cone, 597. 
Of pyramid, 597. 

Of prism, 511. 

Of pyramid, 595. 

Of sphere, 518, 521, 605. 

Taxes, 617. 

Vermont method, 617. 
Term of discount, xix, xx. 
Time between dates, 346, 348, 
609, xiii. 
Principal, interest, and rate 
given, 415, xvii. 
Discount, 424. 
Time draft, cost of, 451, xx. 
Face of, xx. 
Transposing, 377. 


640 


Rules, Continued. 

True discount, 460. 

Triangle, area, 495. 

Volume of cone, 514, 600, 604. 
Of cylinder, 514, 604. 
Of frustum of cone, 603. 
Of frustum of pyramid, 602. 
Of prism, 514, 604. 
Of pyramid, 514, 600, 604. 
Of sphere, 521, 605. 


Scealene triangle, 366, 566. 
Altitude of, 556, 569, 578, 584. 
Area of, 495, 503, 504, 508. 
Construction of, 366, 564, 576, 

578, 579, 580, 582, 584, 585, 
586. 

Secant, 582. 

Second power, 463, xxiii. 

Section (U.S. lands), 524. 

Sector, area of, 516, 592. 
Definition, 572. 

Segment, area of, 516. 

Definition, 516, 572. 

Sextant, 572. 

Share, xxiv. 

Short methods, addition and sub- 

traction, 234, 302, 429. 

Addition of 99, 999, etc., 
396. 

Bank discount, 420. 
Commercial discount, 421, 422. 
Compound interest, 445, 446. 
Division, 233, 268, 341, 400, 430. 

Divisors 125, 25, 75, etc., 
229, 260, 298, 337, 396, 426. 

Divisors ending in ciphers, 
281, 294. ° 

Multiplication, 305, 428, 429,471. 

Fractional multipliers, 399, 
426, 429, 471. 

Multiplier a mixed number, 
323, 340, 363, 399, 426, 429. 

99, 999, etc., as multipliers, 
340, 399, 429, 471. 

125, 25, '75, etc., as multi- 
pliers, 229, 260, 276, 298, 308, 
337, 340, 396, 399, 429, 471. 

Simple interest, 419. 
Subtraction and addition, 234, 
302, 429. 


INDEX. 


Short methods, Continued. 
Subtraction of 99, 999, etc., 
396. 

Sight drafts, 448, 449, 452, 485, 
XXxvi. 

Cost of, 448, 449, 452, 485, xx. 
Face of, 448, 449, 485, xx. 

Sight exercises, 206, 213, 221, 222, 
224, 225, 226, 229, 232, 238, 
248, 244, 251, 252, 254, 260, 
262, 263, 275, 276, 277, 278, 
281, 296, 303, 306, 307, 369, 
370, 374, S577, 381, 426, 442, 
467, 474, 505. 

Similar triangles, 586. 

Slant height, of cone, 512, 5138, 595, 
596. 

Of pyramid, irregular, 596. 
Regular, 512, 594, 596. 

Small circle, 598. 

Solid contents (see Volumes), 358, 
359, 360, 362, 390, 402, 413, 
414, 434, 4385, 442, 461, 462, 
493, 507, 513, 514, 521, 522, 
527, 528, 599, 600, 601, 602, 
603, 604, 605. 

Special drills, 260, 298, 337, 396, 
426, 474. 

Sphere, great circle of, 518, 598. 

Small circle of, 598. 
Surface of, 518, 519, 605. 
Volume of, 521, 522, 605. 
Square, area of, 348, 355, 367, 435, 
461, 503, 504,516. —.- 
Construction of, 569, 578, 574, 
576, 577, 578, 579, 588. 
Definition of, 568. 
Diagonal of, 322, 491, 503, 504. 
Side of, 270, 401, 485, 486. 

Square measure, 354, 607. 

Square root, of algebraic quanti- 
ties, 652, 653. 

Of decimals, 481, 482, 489. 

Of fractions, 473. 

Of integers, 270, 465, 481, 489. 

Of mixed numbers, 478. 

Rules and definitions, 464, 465, 
Xxill, xxiv; 

Stocks and bonds, 888, 442, 443, 
444, 445, 446, 488, 489. 

Definitions, xxv. 


INDEX. 


Subtraction, algebraic quantities, 
532, 533, 534. 
Removing parentheses, 534. 

Decimals, 242, 283, 2938, 364, 440, 
441. 

Denominate numbers, 208, 285, 
287, 288, 312, 331, 332, 335, 
336, 348, 358, 430. 

Fractions, 205, 206, 207, 250, 
256, 262, 263, 277, 296, 310, 
324, 363, 364, 482, 433, 488, 
484. 

Integers, 217, 284, 302, 429. 
Cross-subtraction, 217, 339. 
Principles, definitions, and 

rules, ii. 
Subtrahend, 235, ii. 
Successive discounts (see Com- 
mercial discount), xiv. 
Sum, 285, ii. 
Surface (see Areas and Measure- 
ments). 

Of cone, 512, 
598. 

Of cylinder, 511, 512, 598, 595. 

Of fence, 289, 343, 354, 390, 400, 
435. 

Of frustum of cone, 597. 

Of pyramid, 596, 597. 

Of parallelopipedon, 290, 316, 
345, 854, 406, 418, 414, 434. 

Of prism, 356, 511, 512, 595. 

Of pyramid, 512, 595, 596. 

Of sphere, 518, 519, 599, 605. 

Of walls of room, 289, 315, 316, 
344, 345, 389, 3890, 400, 484, 
461, 462. 

Surveyors’ measure, 607. 


518, 595, 596, 


Tables, apothecaries’ weight, 361, 
607. 
Avoirdupois weight, 227, 361, 
406, 608. 

Long ton, 341, 608. 
Circular measure, 450, 558, 608. 
Cubic measure, 361, 607. 

Dry measure, 227, 607. 
English money, 408, 608. 
Liquid measure, 227, 607. 
Long measure, 285, 607. 
Square measure, 354, 607. 


641 


Tables, Continued. 
Surveyors’ measure, 607. 
Time measure, 226, 608. 
Troy weight, 361, 607. 
U. S. money, 608. 
Tangent, circle, 572, 584. 
Line, 582. 
Taxes, 317, 385, 444, 446, 510, 617, 
618. 
Definitions, xvi. 
Poll, 617, 618. 
Vermont method, 617, 618. 
Term of discount, 402, 403, 404, 
421, 424, 470, xix. 
Third power, 519, xxiii. 
Time drafts, 451. 
Acceptance of, 451, xxvi. 
Cost of, 451, 452, 485. 
Definitions, xx, xxvi. 
Face of, 452, 485. 
Time measure, 226. 
Time note (see Notes), xxv. 
Township (U. S. lands), 524. 
Transposing, 377. 
Trapezium, area of, 368, 413, 495, 
504, 508, 571. 
Construction of, 570, 571. 
Definition, 578. 
Diagonal of, 495, 571. 
Trapezoid, area of, 355, 367, 368, 
4138, 414, 484, 490, 491, 504, 
571. 
Construction of, 569, 570, 571. 
Definition, 568. 
Perimeter of, 491. 
Sides of, 434, 490, 491. 
Triangle, acute-angled, 365, 565, 
570, 571, 584, 585. 
Altitude of, 584, 585. 
Area of, 411, 412, 427, 584, 
585. 
Construction of, 865, 565, 
584. 
Isosceles, 366, 566. 
Altitude of, 495, 567, 578, 
584, 594. 
Area of, 367, 411, 412, 495, 
496, 503. 
Construction of, 366, 566, 
567, 575, 576, 578, 579, 582, 
584. 


642 


Triangle, Continued. 
Oblique-angled, 565. 

Area of, 367, 411, 412, 461, 

495, 496, 503, 556. 
Obtuse-angled, 365, 565. 

Altitude of, 569, 578, 584. 

Area of, 412, 570, 571, 584, 
585. 

Construction of, 571, 583, 
584. 

Scalene, 366, 566. 

Altitude of, 556, 569, 578, 
584. 

Construction of, 564, 575, 
578, 579, 580, 582, 584, 585, 
586. 

Right-angled, 322, 365, 565. 

Area of, 322, 328, 355, 356, 
367, 411, 412, 483, 490, 496, 
526, 551, 555. 

Construction of, 564, 576, 
578, 579, 580, 582, 584, 585, 
586. 

Hypotenuse of, 486, 490, 491, 
496, 506, 525, 526. 

Perimeter of, 491. 

Perpendicular or base of, 


INDEX. 


Triangle, Continued. 
322, 438, 490, 555, 556, 580, 
581, 596. 
True discount, 460. 


United States lands, 524. 

U. 8. rule, partial payments, 458, 
459, 460, xix. 

Usury (see Manual), 610. 


Value of foreign coins, 619. 
Vermont rule, partial payments, 
615, 616, 617. 
Taxes, 617, 618. 
Vertical, angles, 657. 

Lines, 660. 

Volume of cone, 513, 514, 600, 604. 

Of cylinder, 518, 514, 522, 599, 
604. 

Of frustum of cone, 603. 

Of pyramid, 601, 602. 

Of parallelopipedon, 358, 359, . 
360, 362, 390, 402, 413, 414, 
434, 485, 454, 461, 462, 4938. 

Of prism, 513, 514, 599, 600. 

Of pyramid, 513, 600, 601, 604. 

Of sphere, 521, 522, 605. 


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primary difficulties of spelling and sound. 30 cts. 


Sever’s Progressive Speller. For use in advanced primary, intermediate, and gram- 
mar grades. Gives spelling, pronunciation, definition, and use of words. 30 cts. 


Badlam’s Suggestive Lessons in Language. Being Part I and Appendix of 
Suggestive Lessons in Language and Reading. so cts. 


Smith’s Studies in Nature, and Language Lessons. A combination of object 
lessons with language work. socts. Part I bound separately, 25 cts. 


Meiklejohn’s English Language. Treats salient features with a master’s skill and 
with the utmost clearness and simplicity. $1.30. 


Meiklejohn’s English Grammar. Also composition, versification, paraphrasing, etc. 
For high schools and colleges. go cts. 


Meiklejohn’s History of the English Language. 78 pages. Part III of Eng- 
lish Language above, 35 cts. 


Williams’s Composition and Rhetoric by Practice. For high school and col- 
lege. Combines the smallest amount of theory with an abundance of practice. Revised 
edition. $1.00. 


Strang’s Exercises in English. Examples in Syntax, Accidence, and Style foi 
criticism and correction. 50 cts. 


Huffcutt’s English in the Preparatory School. Presents advanced methods 
of teaching English grammar and compositon in the secondary schools. 25 cts. 


Woodward’s Study of English. From primary school to college. 25 cts. 
Genung’s Study of Rhetoric. Shows the most practical discipline. 25 cts. 


See also our list of books for the study of English Literature. 


D. C. HEATH & CO., PUBLISHERS, 
BOSTON. NEW YORK. CHICAGO. 


ELEMENTARY SCIENCE. 


Bailey’s Grammar School Physics. A series of inductive lessons in the elements _ 
of the science. Illustrated. 60 cts. 

Ballard’s The World of Matter. A guide tc the study of chemistry and mineralogy; 
adapted to the general reader, for use as a text-book or as a guide to the teacher in giving 
object-lessons. 264 pages. Illustrated. 1.00. 


Clark’s Practical Methods in Microscopy. Gives in detail descriptions of methods 
that will lead the careful worker to successful results. 233 pages. Illustrated. $1.60. 


Clarke’s Astronomical Lantern. Intended to familiarize students with the constella- 
tions by comparing them with fac-similes on the lantern face. With seventeen slides, 
giving twenty-two constellations. $4 50. 


Clarke’s How to find the Stars. Accompanies the above and helps to an acquaintance 
with the constellations. 47 pages. Paper. 15 cts. 


Guides for Science Teaching. Teachers’ aidsin the instruction of Natural History 
classes in the lower grades. 
. Hyatt’s About Pebbles. 26 pages. Paper. 10 cts. 
II. Goodale’s A Few Common Plants. 61 pages. Paper. 20 cts. 
III. Hyatt’s Commercial and other Sponges. Illustrated. 43 pages. Paper. 20 cts. 
IV. Agassiz’s First Lessons in Natural History. Illustrated. 64 pages. Paper. 
25 cts. 
Vz. Hyatt’s Corals and Echinoderms. Illustrated. 32 pages. Paper. 30 cts. 
VI. Hyatt’s Mollusca. Illustrated. 65 pages. Paper. 30 cts. 
VII. Hyatt’s Worms and Crustacea. Illustrated. 68 pages. Paper. 30 cts. 
VIII. Hyatt’s Insecta. Illustrated. 324 pages. Cloth. $1.25. 
XII. Crosby’s Common Minerals and Rocks. Illustrated. 200 pages. Paper, 40 
cts. Cloth, 60 cts. ;. 
XIII. Richard’s First Lessons in Minerals. 50 pages. Paper. ro cts. 


XIV. Bowditch’s Physiology. 58 pages. Paper. 2octs. 
XV. Clapp’s 36 Observation Lessons in Minerals. 80 pages. Paper. 30 cts. 


XVI. Phenix’s Lessons in Chemistry. 20 cts. 
Pupils’? Note-Book to accompany No. 15. 10 cts. 
Rice’s Science Teaching in the School. With a course of instruction in science. 
for the lower grades. 46 pages. Paper. 25 cts. 
Ricks’s Natural History Object Lessons, Supplies information on plants and 
their products, on animals and their uses, and gives specimen lessons. Fully illustrated. 
332 pages. $1.50. 
Ricks’s Object Lessons and How to Give them. 


Volume I. Gives lessons for primary grades. 200 pages. 9o cts. 
Volume II. Gives lessons for grammar and intermediate grades. 212 pages. 90 cts. 


Shaler’s First Book in Geology. For high school, or highest class in grammar school. 
272 pages. Illustrated. $1.00. 


Shaler’s Teacher’s Methods in Geology. An aid to the teacher of Geology. 
74 pages. Paper. 25 cts. 


Smith’s Studies in Nature. A combination of natural history lessons and language 
work, 48 pages. Paper. 15 cts. 


Sent by mail postpaid on receipt of price. See also our list of books in Science. 


D. C. HEATH & CO., PUBLISHERS, 
BOSTON. NEW YORK. CHICAGO. 


DRAWING AND MANUAL TRAINING. 


Anthony’s Mechanical Drawing. 08 pages of text, and 32 folding plates. $1.50. 
Anthony’s Machine Drawing. 50 pages of text, and 1s folding plates. $1.25. 
Daniels’ Freehand Lettering. 34 pages of text, and 13 folding plates. 8s5 cts. 


Lunt’s Brushwork for Kindergarten and Primary School. 18 lesson-cards 
in colors, with teacher’s pamphlet, in envelope. 30 cts. 


Johnson’s Progressive Lessons in Needlework. Explains needlework from its 
rudiments and gives with illustrations full directions for work during six grades. 117 
pages. Square 8vo. Cloth, $1.00, Boards, 60 cts. 


Seidel’s Industrial Instruction (Smith). A refutation of all objections raised against 
industrial instruction. 170 pages. go cts. 


Thompson’s Educational and Industrial Drawing. 


Primary Free-Hand Series (Nos. 1-4). Each No., per doz., $1.00. 
Primary Free-Hand Manual. 114 pages. Paper. 40 cts. 

Advanced Free-Hand Series (Nos. 5-8). Each No., per doz., $1.50, 
Modei and Object Series (Nos. 1-3). Each No., per doz., $1.75. 
Model and Object Manual. 84 pages. Paper. 35 cts. 

Esthetic Series (Nos. 1-6). Each No., per doz., $1.50. 

fésthetic Manual. 174 pages. Paper. 60 cts. 

Mechanical Series (Nos. 1-6). Each No., per doz., $2.00. 
Mechanical Manual. 172 pages. Paper. 75 cts. 


Thompson’s Manual Training, No. 1. Treats of Clay Modelling, Stick and 
Tablet Laying, Paper Folding and Cutting, Color, and Construction of Geometrical 
Solids. Illustrated. 66 pages. Large 8vo. Paper. 30 cts. 


Thompson’s Manual Training, No. 2. Treats of Mechanical Drawing, Clay- 
Modelling in Relief, Color, Wood Carving, Paper Cutting and Pasting. Illustrated. 
7o pp. Large 8vo. Paper. 30 cts. 


Waldo’s Descriptive Geometry. A large number of problems systematically ar 
ranged, with suggestions. 85 pages. go Cts. 


Whitaker’ s How to Use Wood Working Tools. Lessons in the uses of the 
universal tools: the hammer, knife, plane, rule, chalk-line, square, gauge, chisel, saw, 
and auger. 104 pages. 60 cts. 


Woodward’s Manual Training School. Its aims, methods, and results; with 
detailed courses of instruction in shop-work. Fully illustrated. 374 pages. Octavo. $2.00. 


Sent postpaid by mail on recetpt of price. 


D. C. HEATH & CO., PUBLISHERS, 
BOSTON. NEW YORK. CHICAGO. 


GEOGRAPHY AND MAPS. 


Heath’s Outline Map of the United States. Invaluable for marking territorial 


growth and for the graphic representation of all geographical and historical matter. Small 
(desk) size, 2 cents each; $1.50 per hundred. Intermediate size, 30 cents each. Large. 
size, 50 cts. 


Historical Outline Map of Europe. 12x 18 inches, on bond paper, in black outline. 
3 cents each; per hundred, $2.25. 


Jackson’s Astronomical Geography. Simple enough for grammar schools. Used 
for a bricf course in high school. 40 cts. 


Map of Ancient History. Outline for recording historical growth and statistics (14x 
17 in.), 3 cents each; per 100, $2.25. 


Nichols’ Topics in Geography. A guide for pupils’ use from the primary through 
the eighth grade. 65 cts. 


Picturesque Geography. 12 lithograph plates, 15 x 20 inches, and pamphlet describing 
their use. Per set, $3.00; mounted, $5.00. 


Progressive Outline Maps: United States, *World on Mercator’s Projection (12 x 
20 in.) ; North America, South America, Europe, *Central and Western Europe, Africa, 
Asia, Australia, *British Isles, *England, *Greece, *Italy, New England, Middle Atlan- 
tic States, Southern States, Southern States— western section, Central Eastern States, 
Central Western States, Pacific States, New York, Ohio, The Great Lakes, Washington 
(State), *Palestine (each 10 x 12 in.). For the graphic representation by the pupil of 
geography, geology, history, meteorology, economics, and statistics of all kinds. 2 cents 
each; perhundred, $1.50. 


Those marked with Star (*) are also printed in black outline for use in teaching history. 


Redway’s Manual of Geography. I. Hints to Teachers; II. Modern Facts and 
Ancient Fancies, 65 cts. 


Redway’s Reproduction of Geographical Forms. I. Sand and Clay-Modelling; 
II. Map Drawing and Projection. Paper. 30 cts. 


Roney’s Student’s Outline Map of England. For use in English History and 
Literature, to be filled in by pupils. 5 cts. 


Trotter’s Lessons in the New Geography. Treats geography from the human 
point of view. Adapted for use as a text-book or asa reader. $1.00 


D. C. HEATH & CO., PUBLISHERS, 
BOSTON. NEW YORK. CHICAGO. 


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